# Hard Lefschetz theorem and Hodge–Riemann relations for convex valuations

Andreas Bernig  ,  Jan Kotrbatý  and  Thomas Wannerer  Institut für
Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60054
Frankfurt, Germany<EMAIL_ADDRESS>Charles University, Faculty
of Mathematics and Physics, Sokolovská 49/83, 186 00 Prague, Czechia
<EMAIL_ADDRESS>Friedrich-Schiller-Universität Jena, Fakultät für
Mathematik und Informatik, Institut für Mathematik, Ernst-Abbe-Platz 2, 07743
Jena, Germany<EMAIL_ADDRESS>
###### Abstract.

The algebra of smooth translation-invariant valuations on convex bodies,
introduced by S. Alesker in the early 2000s, was in part proved and in part
conjectured to satisfy properties formally analogous to those of the
cohomology ring of a compact Kähler manifold: Poincaré duality, the hard
Lefschetz theorem, and the Hodge–Riemann relations. Our main result
establishes the hard Lefschetz theorem and the Hodge–Riemann relations in full
generality. As a consequence, we obtain McMullen’s quadratic inequalities,
which are valid for strongly isomorphic polytopes and known to fail in
general, for convex bodies with smooth and strictly positively curved
boundary. Our proof is based on elliptic operator theory and on perturbation
theory applied to unbounded operators on a natural Hilbert space completion of
the space of smooth translation-invariant valuations.

###### 2020 Mathematics Subject Classification:

52A40, 52A39, 52B45, 43A75

AB was supported by DFG grant BE 2484/10-1.

JK was supported by DFG grant BE 2484/5-2 and by Charles University grant
PRIMUS/24/SCI/009.

TW was supported by DFG grant WA 3510/3-1.

##  1\. Introduction

###  1.1. Geometric inequalities

An extremely fruitful approach to the classical isoperimetric problem was
discovered by Minkowski at the end of the nineteenth century. In its core lies
the observation that the isoperimetric inequality can be formulated as an
inequality between mixed volumes of convex bodies. This perspective has led to
striking generalizations and a vast array of related results, yet it seems far
from exhausted. In this respect, we prove here a collection of new
inequalities for mixed volumes that in particular broadly generalizes the
isoperimetric inequality.

The mixed volume is the polarization of the Lebesgue measure under the
Minkowski addition on the space $\mathcal{K}(\RR^{n})$ of convex compact sets
in the $n$-dimensional euclidean space, i.e., the (unique) symmetric,
Minkowski-multilinear function $V:\mathcal{K}(\RR^{n})^{n}\to\RR$ such that
$V(A,\dots,A)=\vol(A)$. It is non-negative and satisfies for all
$A_{1},\dots,A_{n}\in\mathcal{K}(\RR^{n})$ the fundamental Alexandrov–Fenchel
inequality

$\displaystyle V(A_{1},A_{2},A_{3},\dots,A_{n})^{2}\geq
V(A_{1},A_{1},A_{3},\dots,A_{n})V(A_{2},A_{2},A_{3},\dots,A_{n}).$

The Alexandrov–Fenchel inequality is of central importance in convex geometry
and, in fact, far beyond. It subsumes many important inequalities for convex
bodies, such as the Minkowski inequalities and the Brunn–Minkowski inequality.
The two original proofs by Alexandrov [Aleksandrov:Theorie2,
Aleksandrov:Theorie4] and various subsequent alternative proofs
[KavehKhovanskii, Cordero:OneMore, Gromov:ConvexSets, Wang:AF, Teissier:AF,
BraendenLeake:LorentzianCones] revealed its deep, unexpected connections to
different areas, in particular to algebraic and Kähler geometry. Let us also
mention the recent breakthrough due to Shenfeld and van Handel
[ShenfeldHandel:Extremals, ShenfeldHandel:Minkowski] in the open problem of
characterizing the equality cases (see also [Schneider:AF,
Schneider:AFzonoids, Schneider:BM, HugReichert:Support,
HugReichert:Extremizers] for conjectures and partial results). Last but not
least, the Alexandrov–Fenchel inequality is intimately related to
combinatorics [Stanley:TwoApplications, MaShenfeld:Extremals]; for a broader
perspective on this connection see also §1.2 below.

Alexandrov’s first proof of the Alexandrov–Fenchel inequality was based on
approximating convex bodies by strongly isomorphic simple polytopes. This
approach was later extended in a seminal paper of McMullen
[McMullen:SimplePolytopes] where the following much more general result was
proved (see also [Timorin:Analogue, vanHandel:Shepard]):

######  Theorem 1.1 (McMullen [McMullen:SimplePolytopes]).

Let $0\leq k\leq\frac{n}{2}$, $N\in\NN$, $x_{i}\in\RR$, and let
$A_{j}^{i},C_{l}\in\mathcal{K}(\RR^{n})$ be strongly isomorphic simple
polytopes. If

$\displaystyle\sum_{i=1}^{N}x_{i}V(A_{1}^{i},\dots,A_{k}^{i},C_{0},\dots
C_{n-2k},\Cdot[k-1])=0,$

then

$\displaystyle(-1)^{k}\sum_{i,j=1}^{N}x_{i}x_{j}V(A_{1}^{i},\dots,A_{k}^{i},A_{1}^{j},\dots,A_{k}^{j},C_{1},\dots,C_{n-2k})\geq
0$

with equality if and only if
$\sum_{i=1}^{N}x_{i}V(A_{1}^{i},\dots,A_{k}^{i},\Cdot[n-k])=0$.

The two equations in the statement of the theorem are to be understood as
between functions on polytopes, $[k]$ stands for plugging in a polytope $k$
times. A mixed volume with more than $n$ arguments is zero by convention.

The case $k=0$ of Theorem 1.1 is equivalent to the non-negativity of the mixed
volume, which is a well-known, yet non-trivial fact. For $k=1$, the theorem
yields the Alexandrov–Fenchel inequality. This is less obvious but still easy
to see, cf. [vanHandel:Shepard] or [Kotrbaty:HR, Corollary 8.1]. Since these
two inequalities extend to _all_ convex bodies by continuity, it is tempting
to expect the conclusion of Theorem 1.1 to hold without any assumption on
$A_{j}^{i},C_{l}\in\mathcal{K}(\RR^{n})$. However, a naïve approximation
argument does not apply to $k\geq 2$ and, in fact, a recent counterexample due
to van Handel [vanHandel:Shepard] shows that the statement is _wrong_ for
general convex bodies. A natural question therefore is whether McMullen’s
quadratic inequalities possibly hold for subclasses of convex bodies other
than strongly isomorphic simple polytopes.

Inspired by Alexandrov’s second proof of the Alexandrov–Fenchel inequality, we
answer this question affirmatively by proving an analogue of Theorem 1.1 for
sufficiently smooth convex bodies. Namely, denoting the class of convex bodies
with smooth and strictly positively curved boundary by
$\mathcal{K}^{\infty}_{+}(\RR^{n})$, we prove the following theorem. Let
$0\leq k\leq\frac{n}{2}$, $N\in\NN$, $x_{i}\in\RR$, and
$A_{j}^{i},C_{l}\in\mathcal{K}^{\infty}_{+}(\RR^{n})$. If

$\displaystyle\sum_{i=1}^{N}x_{i}V(A_{1}^{i},\dots,A_{k}^{i},C_{0},\dots
C_{n-2k},\Cdot[k-1])=0,$

then

$\displaystyle(-1)^{k}\sum_{i,j=1}^{N}x_{i}x_{j}V(A_{1}^{i},\dots,A_{k}^{i},A_{1}^{j},\dots,A_{k}^{j},C_{1},\dots,C_{n-2k})\geq
0$

with equality if and only if
$\sum_{i=1}^{N}x_{i}V(A_{1}^{i},\dots,A_{k}^{i},\Cdot[n-k])=0$. The conclusion
of Theorem 1.1 in fact holds under the weaker assumption that the support
functions of the bodies $A_{j}^{i}\in\mathcal{K}(\RR^{n})$ are of class
$C^{1,1}$, see Corollary 6.17.

###  1.2. Hodge theory for smooth valuations

Although the statement of Theorem 1.1 is purely convex geometric, there is a
remarkable algebraic structure hidden in the background. In fact, the
inequalities can be, and were originally formulated in
[McMullen:SimplePolytopes] in terms of McMullen’s polytope algebra, more
precisely as the Hodge–Riemann relations for the subalgebra $\Pi(P)$ generated
by polytopes strongly isomorphic to a given simple polytope $P$. For more
details about the polytope algebra, the reader is referred to
[McMullen:PolytopeAlgebra, BernigFaifman:Polytopealgebra, Brion:Polytope]. Let
us only mention here that the product in $\Pi(P)$ is induced by the Minkowski
sum.

An analytic counterpart to McMullens’s combinatorial theory was discovered in
the 2000s through the seminal work of Alesker [Alesker:Irreducibility,
Alesker:Product]. Alesker’s work along with many subsequent developments, in
particular the paper [BernigFu:Convolution] by the first-named author and Fu,
revealed that the Minkowski sum, when restricted to convex bodies of class
$\mathcal{K}^{\infty}_{+}(\RR^{n})$, gives rise to an algebra $\Val^{\infty}$
of smooth valuations. The algebra $\Val^{\infty}$ exhibits numerous properties
analogous to $\Pi(P)$ and even more structure, some of which is reviewed in
§1.3 below. It satisfies Poincaré duality which turned out to have striking
implications in integral geometry [Fu:Unitary, BernigFu:Hig,
BernigSolanes:Kinematic, KotrbatyWannerer:O2, Bernig:SU]. More recently, the
investigation of the connection between the algebra of valuations and
geometric inequalities was initiated [Kotrbaty:HR, KotrbatyWannerer:MixedHR,
KotrbatyWannerer:Harmonic, Alesker:Kotrbaty].

In our paper, we introduce new techniques that allow us to pursue this study
far beyond the previous partial results and complete the analogy between
$\Pi(P)$ and $\Val^{\infty}$. More precisely, we establish new properties of
the algebra of smooth valuations that in particular imply Theorem 1.1.

To formulate our result, recall that a valuation is a function
$\phi:\mathcal{K}(\RR^{n})\to\CC$ satisfying

$\phi(K\cup L)=\phi(K)+\phi(L)-\phi(K\cap L)$

whenever $K,L,K\cup L\in\mathcal{K}(\RR^{n})$. The Banach space of
translation-invariant, continuous valuations is denoted by $\Val$. Alesker has
discovered a natural dense subspace $\Val^{\infty}\subset\Val$ of smooth
valuations. We postpone the precise definition to §2.2; let us only mention
here that the function

$\phi(K)=V(A_{1},\dots,A_{k},K[n-k])$

is a smooth valuation for any fixed
$A_{i}\in\mathcal{K}^{\infty}_{+}(\mathbb{R}^{n})$. In fact, it follows from a
deep theorem of Alesker [Alesker:Irreducibility] together with a recent
observation made independently by Knoerr [Knoerr:MV] and van Handel that such
valuations span $\Val^{\infty}$. Equivalently, the same holds for the
valuations $\vol(\Cdot+A)$, $A\in\mathcal{K}_{+}^{\infty}(\RR^{n})$. The space
of smooth valuations decomposes as

$\displaystyle\Val^{\infty}=\bigoplus_{k=0}^{n}\Val_{k}^{\infty},$

where $\Val_{k}^{\infty}\subset\Val^{\infty}$ is the subspace of
$k$-homogeneous valuations. The convolution product defined by

$\vol(\Cdot+A_{1})*\vol(\Cdot+A_{2})=\vol(\Cdot+A_{1}+A_{2})$

is continuous in the natural Fréchet space topology on $\Val^{\infty}$ and
turns it into a commutative, associative, graded algebra with Poincaré
duality.

For $C\in\mathcal{K}^{\infty}_{+}$ we define the map
$L_{C}:\Val_{k}^{\infty}\to\Val_{k-1}^{\infty}$ by

$(L_{C}\phi)(K)=\left.\frac{d}{dt}\right|_{t=0}\phi(K+tC).$

Equivalently, $L_{C}$ is given by the convolution product with
$nV(C,\Cdot[n-1])$. For any tuple $\mathbf{C}=(C_{1},\dots,C_{m})$ of such
convex bodies we set $L_{\mathbf{C}}=L_{C_{1}}\circ\cdots\circ L_{C_{m}}$
which equals, up to a positive constant, the convolution with
$V(C_{1},\dots,C_{m},\Cdot[n-m])$. Our main result is:

Let $0\leq k\leq\frac{n}{2}$. Fix a tuple
$\mathbf{C}_{0}=(C_{0},\ldots,C_{n-2k})$ of convex bodies from
$\mathcal{K}^{\infty}_{+}(\RR^{n})$ and denote
$\mathbf{C}=(C_{1},\ldots,C_{n-2k})$.

_Hard Lefschetz theorem._ The map
$L_{\mathbf{C}}:\Val_{n-k}^{\infty}\to\Val_{k}^{\infty}$ is an isomorphism of
topological vector spaces.

_Hodge–Riemann relations._ The sesquilinear form

$q_{\mathbf{C}_{0}}(\phi)=(-1)^{k}\b{\phi}*L_{\mathbf{C}}\phi$

on $\ker L_{\mathbf{C}_{0}}\subset\Val_{n-k}^{\infty}$ is positive definite.

Theorem 1.2 confirms a conjecture of the second-named author [Kotrbaty:HR,
Conjecture D] and generalizes many important partial results, previously
obtained in various works. In the special case when all the bodies $C_{i}$ are
euclidean balls, the hard Lefschetz theorem was proved in [Alesker:HLComplex,
BernigBroecker:Rumin] and the Hodge–Riemann relations in [Kotrbaty:HR,
KotrbatyWannerer:Harmonic]. For arbitrary bodies $C_{i}$, however, Theorem 1.2
was known only for $k\leq 1$ [KotrbatyWannerer:MixedHR]. As observed by van
Handel, the previous partial results implied in particular the full statement
in dimensions $n\leq 4$.

It is a remarkable phenomenon that versions of the hard Lefschetz theorem and
Hodge–Riemann relations hold across a broad spectrum of separate theories.
Indeed, apart from Kähler and algebraic geometry, where statements of this
type acquired their names from, and McMullen’s polytope algebra $\Pi(P)$
mentioned above, they have been proved in various different contexts on the
frontier between geometry and combinatorics [ADH:Lagrangian, AHK:Hodge,
EliasWilliamson:Hodge, HuhWang:Enumeration, Karu:HL, BHMPW:Singular]. All
these recent developments had striking consequences, in particular to
combinatorial inequalities, many of which had been open for decades (see also
the ICM addresses of Huh [Huh:ICM18, Huh:ICM22]).

###  1.3. Our methods

There is a fundamental difference between Theorem 1.2 and the analogous
statement in any other context mentioned above. Namely, to the best of our
knowledge, $\Val^{\infty}$ is the only _infinite-dimensional_ algebra where
the hard Lefschetz theorem and Hodge–Riemann relations are known to hold. This
fact makes the proof somewhat more delicate, in particular, one cannot use the
general approach of Cattani [Cattani:Mixed, Cattani:DescentLemma] and deduce
the general statement directly from the previously established case of
euclidean balls. Moreover, in a finite-dimensional situation, the hard
Lefschetz theorem and Hodge–Riemann relations are usually proven
simultaneously, using an inductive argument. This general scheme again fails
for $\Val^{\infty}$ since in particular the injectivity and surjectivity of
the Lefschetz map $L_{\mathbf{C}}$ are not equivalent.

The infinite-dimensional setting of Theorem 1.2 suggests that one should try
to reformulate the statement in a way that is susceptible to the methods of
functional analysis, even if it thus becomes formally stronger than the
original problem. In this respect, our key observation is that there exist two
natural Hilbert space completions $\mathscr{E}_{k}$ and $\mathscr{F}_{k}$ of
$\Val^{\infty}_{k}$. Their existence is less obvious in the language of mixed
volumes, but naturally suggests itself when describing smooth valuations in
terms of smooth differential forms, see §4.2.

We prove the following statement which directly implies the Hodge–Riemann
relations (Theorem 1.2(b)) and thus Theorem 1.1.

With the notation of Theorem 1.2, the sesquilinear form $q_{\mathbf{C}_{0}}$,
densely defined on $\ker
L_{\mathbf{C}_{0}}\cap\Val^{\infty}_{n-k}\subset\ker(L_{\mathbf{C}_{0}}\colon\mathscr{E}_{n-k}\to\mathscr{E}_{k-1})$,
is closable. Its closure $q^{\mathscr{E}}_{\mathbf{C}_{0}}$ is the quadratic
form of a unique self-adjoint operator $Q_{\mathbf{C}_{0}}^{\mathscr{E}}$ on
$\ker L_{\mathbf{C}_{0}}$. The operator $Q_{\mathbf{C}_{0}}^{\mathscr{E}}$ is
strictly positive, has discrete spectrum, and its eigenvalues have finite
multiplicities.

The basic idea of our proof, which goes back to Hilbert’s proof of the
Brunn–Minkowski inequality and was later generalized by Alexandrov in his
second proof of the Alexandrov–Fenchel inequality, is to continuously deform a
given tuple $\mathbf{C}_{0}$ of smooth convex bodies into euclidean balls,
arguing that positivity of the sesquilinear form $q_{\mathbf{C}_{0}}$ is
preserved during this process. The perturbation theory of unbounded operators
on a Hilbert space, in particular the Kato–Rellich theorem, will provide us
with the desired theoretical framework to bring this approach to fruition.

Our strategy is to first establish the hard Lefschetz theorem for

$L_{\mathbf{C}}\colon\mathscr{E}_{n-k}\to\mathscr{E}_{k}$

and to use it to define a family of (unbounded) operators to which
perturbation theory can be applied. Unlike Hilbert and Alexandrov we cannot
directly work with a natural elliptic differential operator associated to
$q_{\mathbf{C}_{0}}$ and yet elliptic operator theory plays an important role
in our proof. Elliptic estimates are crucial in the proof of the surjectivity
of $L_{\mathbf{C}}\colon\mathscr{E}_{n-k}\to\mathscr{E}_{k}$ and elliptic
regularity allows us to obtain the surjectivity of $L_{\mathbf{C}}$ also for
smooth valuations. An important ingredient, entering our argument at two
steps, are the Hodge–Riemann relations in the context of complex linear
algebra, a result due to Timorin [Timorin:Mixed] and the most elementary
incarnation of the Hodge–Riemann relations. We first use it to prove the
injectivity of $L_{\mathbf{C}}$ and later deduce from it the ellipticity of a
particular second order differential operator.

Let us finally remark that in recent years spectral estimates and the closely
related $L^{2}$ method have played an important role in a number of high
profile results in convex geometry. Let us mention the proof of the (B)
conjecture by Cordero-Erausquin, Fradelizi, and Maurey [EFM:BConjecture], the
spectral gap estimate of Kolesnikov and Milman [KolesnikovMilman:logBM,
KolesnikovMilman:BL] in their work on the log-Brunn–Minkowski inequality
[BLYZ:logBM], the proof of the variance conjecture for unconditional convex
bodies by Klartag [Klartag:BerryEsseen], and the confirmation of the
Gardner–Zvavitch conjecture [KolesnikovLivshyts:GZ, EskenazisMoschidis:GZ].

###  1.4. Organization of the article

In Section 2 we collect the necessary background from the theory of smooth
valuations on convex bodies. In Section 3 we develop the linear algebra of
differential forms on the sphere bundle, connect it with mixed volumes and
smooth valuations, and use it to prove injectivity in the hard Lefschetz
theorem. In Section 4, after recalling the necessary results from elliptic
operator theory, we define the Hilbert space completions of the space of
smooth valuations and prove surjectivity in the hard Lefschetz theorem. In
Section 5 we use perturbation theory and previously established properties of
the Lefschetz isomorphism to reduce Theorem 1.3 to the special case of
euclidean balls. The latter is proved, along with further spectral properties
of the Hodge–Riemann form, in Section 6 where we also relax the assumptions of
Theorem 1.1.

### Acknowledgments

We thank Semyon Alesker, Astrid Berg, Dmitry Faifman, Joe Fu, Ramon van
Handel, and Tobias Weth for useful discussions. The first-named author thanks
the ESI Vienna for hosting a research visit that contributed to the results of
the present paper.

##  2\. Background and notation

In this section we collect for later use definitions and results from the
theory of valuations. For the purposes of this paper it suffices to work in
$\RR^{n}$ with its standard euclidean inner product and orientation. Let us
emphasize however that all concepts we introduce, including the convolution of
valuations, are natural in the sense that they can be formulated in a way that
does not depend on these choices. For a more complete presentation of the
principal results of the subject we refer the reader to [Schneider:BM, Chapter
6] and [Alesker:Kent, AleskerFu:Barcelona].

###  2.1. Translation-invariant continuous valuations

Throughout the article, $B^{n}\subset\RR^{n}$ will denote the euclidean unit
ball and $S^{n-1}=\partial B^{n}$ the euclidean unit sphere.

A convex body is a non-empty convex compact subset of $\RR^{n}$. The set of
convex bodies, denoted by $\mathcal{K}(\RR^{n})$, is a locally compact metric
space when equipped with the Hausdorff metric.

######  Definition 2.1.

A valuation is a function $\phi\colon\mathcal{K}(\RR^{n})\to\CC$ satisfying

$\phi(K\cup L)=\phi(K)+\phi(L)-\phi(K\cap L)$

for any $K,L\in\mathcal{K}(\RR^{n})$ whenever the union $K\cup L$ is convex.

A valuation is called translation-invariant if $\phi(K+x)=\phi(K)$ holds for
all $K\in\mathcal{K}(\RR^{n})$ and $x\in\RR^{n}$. The vector space $\Val$ of
translation-invariant, continuous valuations on $\mathcal{K}(\RR^{n})$ admits
a natural grading by the degree of homogeneity. Namely, denoting by
$\Val_{k}\subset\Val$ the subspace of valuations satisfying
$\phi(tK)=t^{k}\phi(K)$ for $t>0$ and $K\in\mathcal{K}(\RR^{n})$, it is a
fundamental result of McMullen [McMullen:EulerType] that

(1) $\Val=\bigoplus_{k=0}^{n}\Val_{k}.$

An important example of a $k$-homogeneous, translation-invariant continuous
valuation is given by the mixed volume

$\phi(K)=V(A_{1},\ldots,A_{n-k},K[k]),$

where $A_{1},\ldots,A_{n-k}$ are arbitrary fixed convex bodies. The spaces
$\Val_{0}$ and $\Val_{n}$ are one-dimensional; $\Val_{0}$ consists of the
constant valuations, $\Val_{n}$ is spanned by the Lebesgue measure $\vol$. The
latter is a non-trivial fact due to Hadwiger [Hadwiger:Vorlesungen]. The
spaces $\Val_{k}$ for $0<k<n$ are infinite-dimensional. The topology on $\Val$
is the topology of uniform convergence on compact subsets. As observed e.g. in
[Alesker:McMullen], it is a consequence of (1) that this topology is in fact
induced by the Banach space norm

$\|\phi\|=\sup_{K\subset B^{n}}|\phi(K)|.$

###  2.2. Smooth valuations

The theory of valuations on convex bodies, as an established line of research,
goes back to the solution of Hilbert’s 3rd problem by Dehn. In the early
2000s, the work of Alesker completely reshaped this classical subject. At the
center of this transformation is Alesker’s discovery of the rich algebraic
structure of the natural dense subspace of smooth valuations
$\Val^{\infty}\subset\Val$. Below we present only a very narrow, if not to say
distorted, image of the modern theory of valuations, focusing only on what is
strictly necessary for this paper. In particular, we discuss neither the
Alesker product [Alesker:Product], the Fourier transform [Alesker:Fourier,
FW:Fourier], nor other natural operations on valuations.

To define smooth valuations, we need to introduce more terminology. First, the
group $\GL(n)$ acts on the Banach space $\Val$ continuously by

$(g\cdot\phi)(K)=\phi(g^{-1}K).$

Second, the normal cycle of a convex body $K\in\mathcal{K}(\RR^{n})$ is, as a
set,

$\nc(K)=\\{(x,u)\in S\RR^{n}\colon u\text{ outward unit normal at }x\in K\\}.$

Here $S\RR^{n}=\RR^{n}\times S^{n-1}$ is the sphere bundle of $\RR^{n}$. In
fact, $\nc(K)\subset S\RR^{n}$ is an oriented compact Lipschitz submanifold of
dimension $n-1$ and thus defines a current through integration. For each
$\omega\in\Omega^{n-1}(S\RR^{n})$,

(2) $\displaystyle\phi(K)=\int_{\nc(K)}\omega$

is a continuous valuation [Alesker:VMfdsIII]. All differential forms we
consider throughout this article are complex-valued. The cartesian product
structure of the sphere bundle induces a bigrading
$\Omega^{k}(S\RR^{n})=\bigoplus_{i+j=k}\Omega^{i,j}(S\RR^{n})$. The subspaces
of forms invariant under the translations $(x,u)\mapsto(x+y,u)$ are denoted by
$\Omega^{k}(S\RR^{n})^{tr}$ and $\Omega^{i,j}(S\RR^{n})^{tr}$. If
$\omega\in\Omega^{k,n-k-1}(S\RR^{n})^{tr}$, then the valuation (2) is in
$\Val_{k}$. Finally, let
$\mathcal{K}_{+}^{\infty}(\RR^{n})\subset\mathcal{K}(\RR^{n})$ be the class of
convex bodies with $C^{\infty}$-smooth boundary and strictly positive Gauss
curvature.

######  Theorem 2.2.

For $\phi\in\Val$ the following are equivalent:

$\phi$ is a smooth vector for the action of $\GL(n,\RR)$ on $\Val$, i.e., the
map $\GL(n,\RR)\to\Val$, $g\mapsto g\cdot\phi$, is smooth.

There exist $\omega\in\Omega^{n-1}(S\RR^{n})^{tr}$ and $c\in\CC$ such that

(3) $\phi(K)=c\vol(K)+\int_{\nc(K)}\omega,\quad K\in\mathcal{K}(\RR^{n}).$

There exist $k_{j}\in\\{0,\dots,n\\}$, $z_{j}\in\CC$, and
$A^{j}_{i}\in\mathcal{K}_{+}^{\infty}(\RR^{n})$ such that

(4)
$\phi(K)=\sum_{j=1}^{m}z_{j}V(A_{1}^{j},\ldots,A_{n-k_{j}}^{j},K[k_{j}]),\quad
K\in\mathcal{K}(\RR^{n}).$

######  Definition 2.3.

A valuation $\phi\in\Val$ is called _smooth_ if it satisfies one and hence all
of the properties listed in Theorem 2.2.

Alesker [Alesker:Product] originally introduced smooth translation-invariant
valuations via (a). By general representation theory, an immediate consequence
of this definition is that the subspace $\Val^{\infty}\subset\Val$ of smooth
valuations is dense. It is not difficult to see that the valuations satisfying
(b) and (c) are smooth vectors in the sense of (a). The implication
(a)$\implies$(b), on the contrary, is highly non-trivial and follows from
Alesker’s irreducibility theorem [Alesker:Irreducibility] and the
Casselman–Wallach theorem [Casselman:HCmod]. That (b) implies (c) was observed
independently by Knoerr [Knoerr:MV] and van Handel.

Both (a) and (b) can be used to define the topology on $\Val^{\infty}$.
Namely, $\Val^{\infty}$ can be topologized either as a subspace of
$C^{\infty}(\GL(n),\Val)$ or as a quotient of
$\Omega^{n-1}(S\RR^{n})^{tr}\oplus\CC$; the resulting topologies coincide.

The valuation (3) will also be denoted by $[[c,\omega]]$. By the kernel
theorem due to the first-named author and Bröcker [BernigBroecker:Rumin],
$\phi=[[c,\omega]]=0$ if and only if

$\phi(\\{0\\})=0\quad\text{and}\quad D\omega+c\cdot dx_{1}\wedge\cdots\wedge
dx_{n}=0,$

where $D$ is the Rumin differential [BernigBroecker:Rumin, Rumin:Contact]
defined as follows. The contact form on $S\RR^{n}$ is the $1$-form

$\alpha|_{x,u}=\sum_{i=1}^{n}u_{i}dx_{i}.$

A form $\eta\in\Omega(S\RR^{n})$ is called vertical if $\alpha\wedge\eta=0$.If
$\omega\in\Omega^{n-1}(S\RR^{n})$, then there is a unique vertical form
$\eta\in\Omega^{n-1}(S\RR^{n})$ such that $d(\omega+\eta)$ is vertical and we
set $D\omega=d(\omega+\eta)$. We will also need the following fact.

######  Lemma 2.4.

Let $1\leq k\leq n$ and $\tau\in\Omega^{k,n-k}(S\RR^{n})^{tr}$. There are
$c\in\CC$ and $\omega\in\Omega^{k,n-1-k}(S\RR^{n})^{tr}$ such that

(5) $\tau=D\omega+c\cdot dx_{1}\wedge\cdots\wedge dx_{n}$

if and only if

(6)
$d\tau=0,\quad\alpha\wedge\tau=0,\quad\text{and}\quad\int_{S^{n-1}}\tau=0.$

######  Proof 2.5.

It follows directly from the definition of the Rumin differential that $\tau$
satisfies (6) if it has the form (5). Conversely, let $\tau$ satisfy (6). If
$k=n$, then $d\tau=0$ implies $\tau=c\cdot dx_{1}\wedge\cdots\wedge dx_{n}$.
For $1<k<n$ the cohomology group $H^{n-k}(S^{n-1})$ vanishes and hence there
is $\omega\in\Omega^{k,n-k-1}(S\RR^{n})^{tr}$ such that $\tau=d\omega$. The
same holds for $k=1$ since $\int_{S^{n-1}}\tau=0$ by assumption. Since $\tau$
is vertical, one has $d\omega=D\omega$.

Note that by convention the integral condition in (6) is automatically
satisfied for $k>1$. Lemma 2.4 and the kernel theorem imply at once one more
equivalent characterization of smooth translation-invariant valuations, cf.
[AleskerBernig:Product, Section 8].

######  Proposition 2.6.

The map

$\Val^{\infty}\to\\{(\tau,z)\mid\tau\in\Omega^{n}(S\RR^{n})^{tr}\text{
satisfies \eqref{eq:closed_vertical}},z\in\CC\\}$

given by

$\phi=\lcur c,\omega\rcur\mapsto\left(D\omega+cdx_{1}\wedge\cdots\wedge
dx_{n},\phi(\\{0\\})\right)$

is a well-defined isomorphism.

###  2.3. The convolution of smooth valuations

The first-named author and Fu [BernigFu:Convolution] have discovered a
continuous bilinear map $\Val^{\infty}\times\Val^{\infty}\to\Val^{\infty}$
called convolution that is uniquely determined by the following property:

(7) $\vol(\Cdot+A_{1})*\vol(\Cdot+A_{2})=\vol(\Cdot+A_{1}+A_{2}),\quad
A_{1},A_{2}\in\mathcal{K}_{+}^{\infty}(\RR^{n}).$

The convolution gives $\Val^{\infty}$ the structure of a commutative graded
algebra with $\vol$ as identity element. Explicitly,
$\Val_{n-k}^{\infty}*\Val_{n-l}^{\infty}\subset\Val_{n-k-l}^{\infty}$. If
$\mathbf{A}$ is a $k$-tuple and $\mathbf{C}$ an $l$-tuple of convex bodies
from $\mathcal{K}_{+}^{\infty}(\RR^{n})$ with $k+l\leq n$, then (7) yields

(8) $\displaystyle
V(\mathbf{A},\Cdot[n-k])*V(\mathbf{C},\Cdot[n-l])=\frac{(n-k)!(n-l)!}{n!(n-k-l)!}V(\mathbf{A},\mathbf{C},\Cdot[n-k-l]).$

In particular, $V(A_{1},\ldots,A_{k},\Cdot[n-k])$ and
$V(A_{1},\Cdot[n-1])*\cdots*V(A_{k},\Cdot[n-1])$ differ only by a positive
factor.

The following description of the convolution in terms of differential forms
will be crucial for our purposes.

Let $*\colon\largewedge^{k}(\RR^{n})^{*}\to\largewedge^{n-k}(\RR^{n})^{*}$ be
the Hodge star isomorphism given by
$\alpha\wedge*\beta=\langle\alpha,\beta\rangle\vol$. We identify
$\Omega^{k,l}(S\RR^{n})^{tr}=\largewedge^{k}(\RR^{n})^{*}\otimes\Omega^{l}(S^{n-1})$
and define the isomorphism
$*_{1}:\Omega^{k,l}(S\RR^{n})^{tr}\to\Omega^{n-k,l}(S\RR^{n})^{tr}$ by

$*_{1}=(-1)^{\binom{n-k}{2}}\ *\otimes\,\mathrm{id}.$

Note that $d\,*_{1}=(-1)^{n}*_{1}d$.

######  Theorem 2.7 ([BernigFu:Convolution]).

Let $k_{1}+k_{2}\leq n$ and suppose
$\omega_{i}\in\Omega^{n-k_{i},k_{i}-1}(S\RR^{n})^{tr}$ for $i=1,2$ . Then
$\lcur 0,\omega_{1}\rcur*\lcur 0,\omega_{2}\rcur=\lcur 0,\omega\rcur$ with

$\omega=*_{1}^{-1}(*_{1}\omega_{1}\wedge*_{1}D\omega_{2}).$

Moreover,

$*_{1}D\omega=*_{1}D\omega_{1}\wedge*_{1}D\omega_{2}.$

###  2.4. Poincaré duality

The convolution of valuations defines a continuous bilinear form
$\Val^{\infty}\times\Val^{\infty}\to\CC$, called the Poincaré pairing, by

(9) $\langle\phi,\psi\rangle=(\phi*\psi)(\\{0\\}).$

In terms of differential forms, if $\phi_{1}=\lcur
0,\omega_{1}\rcur\in\Val_{k}^{\infty}$ and $\phi_{2}=\lcur
0,\omega_{2}\rcur\in\Val_{n-k}^{\infty}$ have complementary degrees with
$0<k<n$, then

$\omega_{1}\wedge
D\omega_{2}\in\Omega^{n,n-1}(S\RR^{n})^{tr}\simeq\Omega^{n-1}(S^{n-1})$

and

(10) $\langle\phi_{1},\phi_{2}\rangle=(-1)^{k}\int_{S^{n-1}}\omega_{1}\wedge
D\omega_{2},$

see [Wannerer:UnitaryAreaMeasures, Proposition 4.2]. A key property of the
product of valuations is the following

######  Theorem 2.8.

The Poincaré pairing (9) is perfect, i.e., the induced map

(11) $\Val^{\infty}(\RR^{n})\to\Val^{\infty}(\RR^{n})^{*}$

is injective with dense image.

Elements of $\Val^{-\infty}=(\Val^{\infty})^{*}$ are called generalized
translation-invariant valuations, see [AleskerFaifman:Lorentzgroup]. Observe
that for $\phi\in\Val^{-\infty}$ and $K\in\mathcal{K}_{+}^{\infty}(\RR^{n})$
it makes sense to define

(12) $\phi(K)=\langle\phi,\vol(\Cdot+K)\rangle,$

since the equality is valid in the classical sense if we consider
$\phi\in\Val^{\infty}$ as a generalized valuations via the embedding (11).

##  3\. Differential forms on the sphere bundle

The hard Lefschetz theorem and the Hodge–Riemann relations for the algebra
$\Val^{\infty}$ will ultimately be reduced to the Hodge–Riemann relations in
complex linear algebra as proven by Timorin. In this section we develop the
linear algebra of differential forms on the sphere bundle necessary for this
reduction and connect it with smooth translation-invariant valuations, in
particular with mixed volumes. As a first application we will be able to prove
injectivity in the hard Lefschetz theorem.

###  3.1. Complex linear algebra

To begin with, we state Timorin’s result, namely, the analog of Theorem 1.2 in
the context of complex linear algebra.

Let $W$ be a real vector space of dimension $2n$ with a complex structure
$J\colon W\to W$. We denote by
$\largewedge_{\CC}^{k}W=\largewedge^{k}W^{*}\otimes_{\RR}\CC$ the space of
complex-valued $k$-forms on the real vector space $W$. Let
$\largewedge_{\CC}^{1}W^{*}=W^{\prime}\oplus\b{W}^{\prime}$ be the
decomposition of the space of $\RR$-linear maps $W\to\CC$ into the
$\CC$-linear and $\CC$-antilinear maps. Then, denoting
$\largewedge^{(p,q)}W^{*}=\largewedge^{p}W^{\prime}\otimes\largewedge^{q}\b{W}^{\prime}$,
one has

(13) $\largewedge^{k}_{\CC}W^{*}=\bigoplus_{p+q=k}\largewedge^{(p,q)}W^{*}.$

Let us clarify that we violate the standard notation here since in our
applications $W$ will usually be a direct sum of two vector spaces and
$\largewedge^{p,q}W^{*}$ will refer to the bidegree decomposition in this
other context.

A $(1,1)$-form $\omega\in\largewedge^{(1,1)}W^{*}$ is called positive if it is
real, i.e., $\b{\omega}=\omega$, and satisfies $\omega(w,Jw)>0$ for each non-
zero $w\in W$.

######  Theorem 3.1 (Timorin [Timorin:Mixed]).

Let $p+q\leq n$ and let $\omega_{0},\omega_{1},\ldots,\omega_{n-p-q}$ be
positive $(1,1)$-forms. Set
$\Omega=\omega_{1}\wedge\ldots\wedge\omega_{n-p-q}\in\largewedge^{(n-p-q,n-p-q)}W^{*}$.
Then the following holds:

_Hard Lefschetz theorem._ The wedge product with $\Omega$ is an isomorphism

$\Omega:\largewedge^{(p,q)}W^{*}\to\largewedge^{(n-q,n-p)}W^{*}.$

_Hodge–Riemann relations._ Let

$P^{(p,q)}=\\{\omega\in\largewedge^{(p,q)}W^{*}:\omega_{0}\wedge\Omega\wedge\omega=0\\}$

be the subspace of primitive forms. Then the sesquilinear form

$q(\alpha,\beta)=\mathbf{i}^{p-q}(-1)^{\binom{p+q}{2}}\alpha\wedge\b{\beta}\wedge\Omega$

on $P^{(p,q)}$ is positive definite.

Here and in what follows we use the canonical orientation of $W$ to orient
$\largewedge^{2n}W^{*}$ and the notation $\mathbf{i}=\sqrt{-1}$.

###  3.2. The linear algebra of $(p,p)$-forms

We are going to apply the linear Hodge–Riemann relations in the situation
$W=V\oplus V$, where $V$ is an $n$-dimensional real vector space and
$J(v,w)=(-w,v)$. To this end we first need to verify that a certain class of
forms is of type $(p,p)$.

Let
$K^{p}(V)\subset\largewedge_{\CC}^{p}V^{*}\otimes\largewedge_{\CC}^{p}V^{*}$
be the kernel of the map

$\largewedge_{\CC}^{p}V^{*}\otimes\largewedge_{\CC}^{p}V^{*}\to\largewedge^{p-1}_{\CC}V^{*}\otimes\largewedge^{p+1}_{\CC}V^{*}$

given by
$\beta\otimes\gamma\mapsto\sum_{i=1}^{n}\iota_{e_{i}}\beta\otimes\alpha_{i}\wedge\gamma,$
where $e_{1},\ldots,e_{n}$ is a basis of $V$ and
$\alpha_{1},\ldots,\alpha_{n}$ is the dual basis. We need the following fact,
proved for example in [BFSW, Lemma 5.7].

######  Lemma 3.2.

$K^{p}(V)$ is irreducible under the natural action of $\GL(n,\CC)$.

At the same time, we can view $K^{p}(V)$ as a subspace of
$\largewedge_{\CC}^{2p}W^{*}$, where $W=V\oplus V$. In this respect, the
following fact will be important.

######  Proposition 3.3.

$K^{p}(V)\subset\largewedge^{(p,p)}W^{*}.$

######  Proof 3.4.

The decomposition (13) is compatible with the natural action of $\GL(n,\CC)$,
i.e., each summand $\largewedge^{(p,q)}W^{*}$ is an invariant subspace.
Observe that $K^{p}(V)$ and $\largewedge^{(p,p)}W^{*}$ intersect non-
trivially; indeed, if $(x_{i},y_{i})$ are coordinates on $W=V\oplus V$ and
$dz_{i}=dx_{i}+\mathbf{i}dy_{i}$, then

$dz_{1}\wedge\cdots\wedge dz_{p}\wedge d\b{z}_{1}\wedge\cdots\wedge
d\b{z}_{p}=c\ dx_{1}\wedge\cdots dx_{p}\wedge dy_{1}\wedge\cdots\wedge
dy_{p},$

for some $c\neq 0$, lies in the intersection. Since $K^{p}(V)$ is irreducible,
the claim follows.

###  3.3. Smooth valuations and $(p,p)$-forms

Let us write $(x,u)$ for the standard coordinates on $\RR^{n}\times\RR^{n}$.
To each convex body $A\in\mathcal{K}^{\infty}_{+}(\RR^{n})$ we associate its
support function

$h_{A}(u)=\sup_{a\in A}\langle a,u\rangle,\quad u\in\RR^{n},$

and the following smooth differential forms on
$\RR^{n}\times(\RR^{n}\setminus\\{0\\})$:

(14)
$\displaystyle\begin{split}\alpha_{A}&=\sum_{i=1}^{n}\pder[h_{A}]{u_{i}}dx_{i},\\\
\omega_{A}&=h_{A}\iota_{T}\vol,\\\ \Omega_{A}&=-d\alpha_{A},\end{split}$

where $T=\sum_{i=1}^{n}u_{i}\pder{x_{i}}$ denotes the Reeb vector field and
$\vol$ is the volume form on the first factor. By restriction we obtain
differential forms on the sphere bundle $S\RR^{n}=\RR^{n}\times S^{n-1}$ that
we will not distinguish notationally.

######  Lemma 3.5.

Let $A\in\mathcal{K}_{+}^{\infty}(\RR^{n})$. We have

$nV(A,K[n-1])=\int_{\nc(K)}\omega_{A},\quad K\in\mathcal{K}(\RR^{n}),$

and $*_{1}D\omega_{A}=\Omega_{A}$.

######  Proof 3.6.

First, if $P\in\mathcal{K}(\RR^{n})$ is a polytope, then

$nV(P[n-1],A)=\sum_{F}h_{A}(\nu_{F})\vol_{n-1}(F)=\int_{\nc(P)}\omega_{A},$

where the sum is over the facets of $P$ and $\nu_{F}$ denotes the unit facet
normal. The general case of the first statement then follows by continuity.

To prove the second identity, we first show that
$d(\omega_{A}+\alpha\wedge\xi)$ with

$\xi=-\sum_{i=1}^{n}\pder[h_{A}]{u_{i}}\iota_{\pder{x_{i}}}\iota_{T}\vol$

is vertical. We have

$d\omega_{A}=dh_{A}\wedge\iota_{T}\vol+h_{A}d\iota_{T}\vol,$

and the second term is vertical. Since

$d\alpha\wedge\xi=\sum_{k=1}^{n}du_{k}\wedge dx_{k}\wedge\xi=-dh_{A}\wedge
i_{T}\vol$

we obtain

$D\omega_{A}=d(\omega_{A}+\alpha\wedge\xi)=d\omega_{A}+d\alpha\wedge\xi-\alpha\wedge
d\xi=h_{A}d\iota_{T}\vol-\alpha\wedge d\xi.$

Next, we get

$\displaystyle\alpha\wedge d\xi$
$\displaystyle=-\alpha\wedge\sum_{i,k=1}^{n}\frac{\partial^{2}h_{A}}{\partial
u_{i}\partial u_{k}}du_{k}\wedge\iota_{\frac{\partial}{\partial
x_{i}}}\iota_{T}\vol+\alpha\wedge\sum_{i=1}^{n}\frac{\partial h_{A}}{\partial
u_{i}}\iota_{\frac{\partial}{\partial x_{i}}}d\iota_{T}\vol.$

Since $\sum_{i=1}^{n}\pder[h_{A}]{u_{i}}u_{i}=h_{A}$ by homogeneity of
$h_{A}$, the second term is equal to $h_{A}d\iota_{T}\vol$.

Using $\sum_{i=1}^{n}\frac{\partial^{2}h_{A}}{\partial u_{i}\partial
u_{k}}u_{i}=0$ in the third equality, the claimed identity now follows from

$\displaystyle*_{1}D\omega_{A}$
$\displaystyle=*_{1}\left(\alpha\wedge\sum_{i,k=1}^{n}\frac{\partial^{2}h_{A}}{\partial
u_{i}\partial u_{k}}du_{k}\wedge\iota_{\frac{\partial}{\partial
x_{i}}}\iota_{T}\vol\right)$
$\displaystyle=(-1)^{n}\sum_{i,k=1}^{n}\frac{\partial^{2}h_{A}}{\partial
u_{i}\partial u_{k}}*_{1}\left(\alpha\wedge\iota_{\frac{\partial}{\partial
x_{i}}}\iota_{T}\vol\wedge du_{k}\right)$
$\displaystyle=\sum_{i,k=1}^{n}\frac{\partial^{2}h_{A}}{\partial u_{i}\partial
u_{k}}dx_{i}\wedge du_{k}$ $\displaystyle=-d\alpha_{A}$
$\displaystyle=\Omega_{A}.$

######  Corollary 3.7.

Let $A\in\mathcal{K}_{+}^{\infty}(\RR^{n})$. If $\phi=\lcur 0,\omega\rcur$ for
$\omega\in\Omega^{k,n-k-1}(S\RR^{n})^{tr}$, then

$\phi(A)=\frac{(-1)^{\binom{n}{2}}}{k!}\int_{S^{n-1}}\ast_{1}\omega\wedge\Omega_{A}^{k}.$

######  Proof 3.8.

Denote $\psi_{A}=nV(A,\Cdot[n-1])$. By Lemma 3.5 and Theorem 2.7, we have

$\displaystyle
k!\phi(A)=(L_{A}^{k}\phi)(\\{0\\})=\langle\phi,\psi_{A}^{k}\rangle=(-1)^{\binom{n}{2}}\int_{S^{n-1}}\ast_{1}\omega\wedge\Omega_{A}^{k}.$

The sphere bundle of any riemannian manifold has a natural contact structure
given by a distribution of contact planes $H_{x,u}\subset T_{x,u}SM$. In the
special case $M=\RR^{n}$, each contact plane $H_{x,u}=u^{\perp}\oplus
u^{\perp}$ is a complex vector space of dimension $n-1$ with the complex
structure $J(X_{1},X_{2})=(-X_{2},X_{1})$.

We say that a differential form $\omega\in\Omega^{p+q}(S\RR^{n})$ is a
$(p,q)$-form if each $\omega|_{x,u}$ belongs to $\largewedge^{p+q}H^{*}_{x,u}$
and is a $(p,q)$-form in the sense of Subsection 3.1. Note that the first
condition is equivalent to $i_{T}\omega=0$. A $(1,1)$-form
$\omega\in\Omega^{2}(S\RR^{n})$ is called positive if each
$\omega|_{x,u}\in\largewedge^{1,1}H^{*}_{x,u}$ is positive.

######  Proposition 3.9.

If $\omega\in\Omega^{n-k,k-1}(S\RR^{n})^{tr}$, then $*_{1}D\omega$ is a
$(k,k)$-form. Moreover, the $(1,1)$-form $*_{1}D\omega_{A}$ is positive for
each $A\in\mathcal{K}_{+}^{\infty}(\RR^{n})$.

######  Proof 3.10.

First, the identities $\alpha\wedge D\omega=0$ and $d\alpha\wedge D\omega=0$
imply $i_{T}(*_{1}D\omega)=0$. Hence, by Proposition 3.3, $*_{1}D\omega\in
K^{k}(u^{\perp})\subset\largewedge^{(k,k)}H_{x,u}^{*}$. Second, since $A$ has
a strictly positively curved boundary,

$\Omega_{A}(X,JX)=\Hess h_{A}(u)(X_{1},X_{1})+\Hess h_{A}(u)(X_{2},X_{2})>0$

for every non-zero $X=(X_{1},X_{2})\in H_{x,u}$.

###  3.4. Injectivity in the hard Lefschetz theorem

We are now in a position to deduce injectivity in the hard Lefschetz theorem
for smooth valuations from the linear Hodge–Riemann relations.

######  Proposition 3.11.

For each tuple $\mathbf{C}=(C_{1},\ldots,C_{n-2k})$ of convex bodies from
$\mathcal{K}_{+}^{\infty}(\RR^{n})$, the map
$L_{\mathbf{C}}\colon\Val^{\infty}_{n-k}\to\Val^{\infty}_{k}$ is injective.

######  Proof 3.12.

For $k=0$, this follows trivially from (8) and $V(C_{1},\ldots,C_{n})\neq 0$.
Let us assume $k>0$ from now on. By translation invariance, we may further
assume that each $C_{i}$ contains the origin in its interior.

Let $\phi=\lcur 0,\omega\rcur$ be in the kernel of $L_{\mathbf{C}}$. Put
$\tau=*_{1}D\omega\in\Omega^{k,k}(S\RR^{n})$. By (8), Theorem 2.7, Lemma 3.5,
and Proposition 3.9,

(15) $\Omega_{C_{1}}\wedge\cdots\wedge\Omega_{C_{n-2k}}\wedge\tau=0.$

Thus each $\tau|_{x,u}$ is a $(k,k)$-form that is primitive with respect to
the positive $(1,1)$-forms
$\Omega_{C_{1}}|_{x,u},\ldots,\Omega_{C_{n-2k}}|_{x,u}$.

The linear Hodge–Riemann relations (Theorem 3.1), for the complex vector space
$(H_{x,u},J)$ of complex dimension $n-1$ and $p=q=k$ yield

$(-1)^{k}\tau\wedge\b{\tau}\wedge\Omega_{C_{1}}\wedge\cdots\wedge\Omega_{C_{n-2k-1}}\geq
0\quad\text{at }(x,u)$

with equality if and only if $\tau|_{x,u}=0$. Each convex body
$A\in\mathcal{K}_{+}^{\infty}(\RR^{n})$ with the origin in the interior
satisfies $\alpha_{A}|_{x,u}(u,0)=h_{A}(u)>0$. We thus have

(16)
$(-1)^{k}\alpha_{C_{n-2k}}\wedge\tau\wedge\b{\tau}\wedge\Omega_{C_{1}}\wedge\cdots\wedge\Omega_{C_{n-2k-1}}\geq
0\quad\text{at }(x,u)$

with equality if and only if $\tau|_{x,u}=0$.

Since $\tau$ is closed and $H^{k}(S^{n-1})=0$, we may write $\tau=d\kappa.$
Using Stokes’ theorem and (15),

$\displaystyle\int_{S^{n-1}}\alpha_{C_{2k-n}}\wedge\tau\wedge\b{\tau}\wedge\Omega_{C_{1}}\wedge\cdots\wedge\Omega_{C_{2k-n-1}}$
$\displaystyle\quad=-\int_{S^{n-1}}\Omega_{C_{2k-n}}\wedge\tau\wedge\b{\kappa}\wedge\Omega_{C_{1}}\wedge\cdots\wedge\Omega_{C_{2k-n-1}}=0.$

This implies that equality holds in (16) for every $(x,u)$ and thus $\tau=0$,
which implies that $\phi=0$.

##  4\. Surjectivity in the hard Lefschetz theorem

Let $\mathbf{C}=(C_{1},\ldots,C_{n-2k})$ be a tuple of convex bodies from
$\mathcal{K}_{+}^{\infty}(\RR^{n})$. The goal of this section is to establish
the surjectivity of
$L_{\mathbf{C}}\colon\Val^{\infty}_{n-k}\to\Val^{\infty}_{k}$, thus proving
the hard Lefschetz theorem. As we have seen in the previous section, after a
careful description of the Lefschetz map in terms of differential forms on the
sphere bundle, the injectivity is an immediate consequence of Timorin’s linear
Hodge–Riemann relations and Stokes’ theorem. Proving surjectivity however will
require tools from PDE theory, in particular the machinery of Sobolev spaces,
weak solutions, and elliptic regularity.

We will introduce in this section two natural Hilbert space completions,
denoted $\mathscr{E}_{k}$ and $\mathscr{F}_{k}$, of the space of smooth
valuations $\Val_{k}^{\infty}$. The existence of these two Hilbert space
completions is a result of the two descriptions of smooth valuations via
smooth differential forms: forms of degree $n-1$ or closed vertical $n$-forms.
In the present section, the Hilbert space $\mathscr{E}_{k}$ will play a lesser
role, but it will be of central importance for our proof of the Hodge–Riemann
relations.

###  4.1. Facts from elliptic operator theory

Let us recall from [LawsonMichelson:Spin] several fundamental results about
elliptic operators.

The Sobolev $s$ norm of a Schwartz function $u\colon\RR^{n}\to\CC^{p}$ is
defined by

(17) $\|u\|_{s}=\int_{\RR^{n}}(1+|\xi|)^{2s}|\hat{u}(\xi)|^{2}d\xi,$

where $\hat{u}$ denotes the Fourier transform of $u$. Note that
$\|\Cdot\|_{0}$ is the $L^{2}$ norm.

Let $E\to X$ be a complex vector bundle over a compact manifold $X$. Let
$U_{i}$, $i=1,\ldots,N$, be an open cover of $X$ by coordinate neighbourhoods
such that $E$ is locally trivial over each $U_{i}$. Fix local coordinates,
local trivializations, and a partition of unity $(\rho_{i})_{i=1}^{N}$
subordinate to $(U_{i})_{i=1}^{N}$. Let $\Gamma(E)$ denote the space of smooth
sections of $E$. Any $u\in\Gamma(E)$ can be written as $u=\sum_{i=1}^{N}u_{i}$
with $u_{i}=\rho_{i}u$. For $s\in\RR$, the Sobolev $s$ norm on $\Gamma(E)$ is
defined by

$\|u\|_{s}=\sum_{i=1}^{N}\|u_{i}\|_{s},$

where, using $E|_{U_{i}}\cong U_{i}\times\CC^{p}$, the norms on the right-hand
side are given by (17). Different choices of the coordinate system or the
local trivializations give equivalent Sobolev norms. The completion of
$\Gamma(E)$ with respect to this norm is denoted $H^{s}(E)$. It is clear from
(17) that $H^{s}(E)$ is a Hilbert space.

Let $E$ and $F$ be complex vector bundles of ranks $p$ and $q$ over $X$. A
linear map $P\colon\Gamma(E)\to\Gamma(F)$ is called a differential operator of
order $m$ if each point of $X$ possesses an open neighbourhood $U$ with
coordinates $x_{1},\ldots,x_{n}$ and a local trivialization $E|_{U}\cong
U\times\CC^{p}$ such that $P$ can be written as

$P=\sum_{|\alpha|\leq m}A^{\alpha}\frac{\partial^{|\alpha|}}{\partial
x^{\alpha}},$

where $A^{\alpha}$ are $q\times p$ matrices of complex-valued functions on $U$
and $A^{\alpha}\neq 0$ for some $\alpha$ with $|\alpha|=m$.

######  Lemma 4.1.

Let $P\colon\Gamma(E)\to\Gamma(F)$ be a differential operator of order $m$
over a compact manifold $X$. For any $s\in\RR$, the differential operator $P$
extends to a bounded operator $P\colon H^{s}(E)\to H^{s-m}(F)$.

The principal symbol of a differential operator $P$ is a section $\sigma(P)$
of $\pi^{*}\Hom(E,F)$, the pullback bundle of $\Hom(E,F)$ under the canonical
projection $\pi\colon T^{*}X\to X$, such that for $\xi=\sum\xi_{k}dx_{k}$,

$\sigma_{\xi}(P)=\mathbf{i}^{m}\sum_{|\alpha|=m}A^{\alpha}(x)\xi^{\alpha},$

where $\mathbf{i}=\sqrt{-1}$. If $P\colon\Gamma(E)\to\Gamma(F)$ and
$Q\colon\Gamma(F)\to\Gamma(G)$ are differential operators, then
$\sigma_{\xi}(Q\circ P)=\sigma_{\xi}(Q)\sigma_{\xi}(P)$. The differential
operator $P$ is called elliptic if for non-zero cotangent vectors $\xi\in
T^{*}X$ the symbol $\sigma_{\xi}(P)\colon E_{x}\to F_{x}$ is invertible for
all $x\in X$. In particular $p=q$ in this case.

######  Theorem 4.2.

Let $P\colon\Gamma(E)\to\Gamma(F)$ be an elliptic operator of order $m$ over a
compact manifold $X$. The following holds:

_Elliptic regularity._ For any open set $U\subset X$ and any $u\in H^{s}(E)$,

$Pu|_{U}\in C^{\infty}\Longrightarrow u|_{U}\in C^{\infty}.$

For each $s$, the kernel of $P\colon H^{s}(E)\to H^{s-m}(F)$ consists of
smooth sections and its dimension is finite and independent of $s$.

_Elliptic estimate._ For each $s$, there is a constant $C$ such that

$\|u\|_{s}\leq C(\|u\|_{s-m}+\|Pu\|_{s-m})$

for all $u\in H^{s}(E)$.

For each $s$, there is a constant $C$ such that

$\|u\|_{s}\leq C\|Pu\|_{s-m}$

for all $u\in(\ker P)^{\perp}$, the orthogonal complement of the kernel of $P$
in $H^{s}(E)$.

######  Proof 4.3.

Items (a), (b), and (c) are contained in [LawsonMichelson:Spin, Theorem 5.2].
A short argument (see, e.g., the proof of Lemma 4.14 in [Wells:Complex]) shows
that the existence of a parametrix implies (d).

A hermitian metric on $E\to X$ and a smooth measure $\mu$ on $X$ define an
$L^{2}$ norm

$\|u\|_{L^{2}}=\left(\int_{X}|u|^{2}d\mu\right)^{1/2}$

on $\Gamma(E)$. If $P\colon\Gamma(E)\to\Gamma(F)$ is a differential operator,
then the adjoint of $P$ is the differential operator
$P^{*}\colon\Gamma(F)\to\Gamma(E)$ characterized by the property

$\langle Pu,v\rangle_{L^{2}}=\langle u,P^{*}v\rangle_{L^{2}}$

for $u\in\Gamma(E)$ and $v\in\Gamma(F)$. The symbol of the adjoint satisfies
$\sigma_{\xi}(P^{*})=\sigma_{\xi}(P)^{*}$.

######  Example 4.4.

Let $X$ be a compact riemannian manifold. The Laplace–Beltrami operator
$\Delta\colon\Omega^{k}(X)\to\Omega^{k}(X)$ is defined by
$\Delta=d\delta+\delta d$, where we use the standard notation $\delta:=d^{*}$.
It is an elliptic differential operator of order $2$ with principal symbol

$\sigma_{\xi}(\Delta)=|\xi|^{2}.$

###  4.2. Hilbert space completions of the space of valuations

Recall that if $Y\subset X$ is a closed subspace, then $X/Y$ is naturally a
normed space with respect to $\|\pi(x)\|=\inf_{y\in Y}\|x+y\|$, $x\in X$,
where $\pi:X\to X/Y$ is the canonical projection. Let $i\colon X\to\wt X$ be
the completion of $X$. By parallelogram law, if $X$ is an inner product space,
then so are $X/Y$ and $\wt X$. The following lemma is well known.

######  Lemma 4.5.

Let $X$ be an inner product space and $Y\subset X$ a closed subspace. Let
$i\colon X\to\wt X$ be the completion of $X$. Then $\wt X/\overline{i(Y)}\cong
i(Y)^{\perp}\subset\wt X$ is the completion of $X/Y$.

Let $E_{i,j}\to S^{n-1}$ denote the complex vector bundle
$\largewedge_{\CC}^{i}(\RR^{n})^{*}\boxtimes\largewedge_{\CC}^{j}T^{*}S^{n-1}$
with the canonical hermitian metric. The space of smooth sections of $E_{i,j}$
is just $\Omega^{i,j}(S\mathbb{R}^{n})^{tr}$. Let us introduce the notation
$\Omega_{H^{s}}^{i,j}(S\RR^{n})^{tr}=H^{s}(E_{i,j})$ and
$\Omega_{L^{2}}^{i,j}(S\RR^{n})^{tr}=L^{2}(E_{i,j})$. Explicitly, the $L^{2}$
inner product on $\Omega_{L^{2}}^{i,j}(S\RR^{n})^{tr}$ is given by

(18)
$\langle\omega_{1},\omega_{2}\rangle_{L^{2}}=\int_{S^{n-1}}\omega_{1}\wedge\overline{*\omega_{2}},\quad\omega_{1},\omega_{2}\in\Omega^{i,j}(S\mathbb{R}^{n})^{tr},$

where $*$ is the Hodge star operator.

Define for $1\leq k\leq n-1$

(19) $\displaystyle\begin{split}X_{k}&=\Omega^{k,n-k-1}(S\RR^{n})^{tr},\\\
Y_{k}&=\\{\omega\in X_{k}\colon D\omega=0\\},\\\
Z_{k}&=\\{\tau\in\Omega^{k,n-k}(S\RR^{n})^{tr}\colon\text{properties
}\eqref{eq:closed_vertical}\text{ hold}\\}.\end{split}$

We consider $X_{k}$ and $Z_{k}$ as subspaces of
$\Omega^{k,n-k-1}_{L^{2}}(S\RR^{n})^{tr}$ and
$\Omega^{k,n-k}_{L^{2}}(S\RR^{n})^{tr}$ with the induced $L^{2}$ inner
product. To see that $Y_{k}$ is closed, just observe that the extension
$D\colon\Omega^{k,n-1-k}_{L^{2}}(S\RR^{n})^{tr}\to\Omega^{k,n-k}_{H^{-2}}(S\RR^{n})^{tr}$
is continuous.

######  Definition 4.6.

For $1\leq k\leq n-1$, we denote by $\mathscr{E}_{k}$ the Hilbert space
completion of $X_{k}/Y_{k}$ and by $\mathscr{F}_{k}$ the Hilbert space
completion of $Z_{k}$, in both cases with respect to the induced $L^{2}$ inner
product. Moreover, we set $\mathscr{E}_{0}=\mathscr{F}_{0}=\Val_{0}$ and
$\mathscr{E}_{n}=\mathscr{F}_{n}=\Val_{n}$, with the norms defined in such a
way that $\|1\|_{\mathscr{E}}=\|1\|_{\mathscr{F}}=1$ and
$\|\vol\|_{\mathscr{E}}=\|\vol\|_{\mathscr{F}}=1$.

######  Example 4.7.

Let us describe these inner products in the special case $k=n-1$ in more
familiar terms. In this degree, every smooth valuation can be expressed as
$\phi(K)=nV(A,K[n-1])-nV(C,K[n-1])$ with some convex bodies
$A,C\in\mathcal{K}_{+}^{\infty}$. Hence by Lemma 3.5

$\|\phi\|_{\mathscr{E}}=\|\omega_{A}-\omega_{C}\|_{X_{n-1}/Y_{n-1}}=\inf_{x\in\RR^{n}}\left(\int_{S^{n-1}}|h_{A+x}-h_{C}|^{2}du\right)^{1/2}$

since $D(\omega_{A}-\omega_{C})=0$ if and only if $h_{A}-h_{C}$ is linear.
Similarly,

$\|\phi\|_{\mathscr{F}}=\|D\omega_{A}-D\omega_{C}\|_{Z_{n-1}}=\left(\int_{S^{n-1}}\sum_{i,j=1}^{n}\left|\frac{\partial^{2}h_{A}}{\partial
u_{i}\partial u_{j}}-\frac{\partial^{2}h_{C}}{\partial u_{i}\partial
u_{j}}\right|^{2}du\right)^{1/2}.$

Our first goal in this section is to prove a version of the Rellich–Kondrachov
theorem. We will need the following lemma.

######  Lemma 4.8.

Let $\tau\in\Omega^{m}_{L^{2}}(S^{n-1})$, $1\leq m\leq n-1$, satisfy $d\tau=0$
or, if $m=n-1$, let $\tau$ be perpendicular to the riemannian volume form.
Then there exists $\omega\in\Omega_{H^{1}}^{m-1}(S^{n-1})$ such that

$d\omega=\tau$ and

$\omega\in\operatorname{im}(\delta\colon\Omega_{H^{2}}^{m}(S^{n-1})\to\Omega_{H^{1}}^{m-1}(S^{n-1}))$.
Moreover, $\omega$ is uniquely determined by these two properties and there
exists a constant $C=C(k,n)$ depending only $k$ and $n$ such that

$\|\omega\|_{1}\leq C\|\tau\|_{0}.$

######  Proof 4.9.

To prove existence observe that we can find
$\sigma\in\Omega_{H^{2}}^{m}(S^{n-1})$ such that $\Delta\sigma=\tau$. Indeed,
if $m<n-1$ then $\Delta$ is a bijection, and if $m=n-1$, the additional
condition on $\tau$ implies that $\tau$ is orthogonal to the kernel of
$\Delta$. By the definition of the Laplace–Beltrami operator it follows that
$\tau=d\omega+\delta\gamma$, where $\omega=\delta\sigma$ and $\gamma=d\sigma$.
Since the exact and co-exact forms are orthogonal,

$\|\delta\gamma\|_{L^{2}}=\langle\delta\gamma,\delta\gamma\rangle_{L^{2}}=\langle\tau,\delta\gamma\rangle_{L^{2}}=\langle
d\tau,\gamma\rangle_{L^{2}}=0.$

Thus $\tau=d\omega$ with
$\omega\in\operatorname{im}(\delta\colon\Omega_{H^{2}}^{m}(S^{n-1})\to\Omega_{H^{1}}^{m-1}(S^{n-1}))$.

To prove uniqueness let $\omega=\delta\sigma$ and suppose $d\omega=0$. Thus
$\omega$ is harmonic and hence vanishes or, if $m=1$, is a constant function.
But constant functions are perpendicular to the image of $\delta$ and hence
$\omega=0$ in all cases.

Finally observe that $\delta\tau=\delta d\omega=\Delta\omega$. If $m>1$, then

$\|\omega\|_{1}\leq C\|\Delta\omega\|_{-1}\leq C\|\tau\|_{0}$

by Theorem 4.2(d). If $m=1$, then $\omega$ is perpendicular to the kernel of
$\Delta$, since $\omega$ is co-exact, and hence the inequalities are valid
also in this case.

######  Proposition 4.10.

We have continuous inclusions

$\Val_{k}^{\infty}\subset\mathscr{F}_{k}\subset\mathscr{E}_{k}\subset\Val_{k}^{-\infty}.$

Moreover, $\mathscr{F}_{k}\hookrightarrow\mathscr{E}_{k}$ is compact.

######  Proof 4.11.

We may assume that $1\leq k\leq n-1$. The $(n-1)$\- and $n$-forms that make up
the spaces $\mathscr{E}_{k}$ and $\mathscr{F}_{k}$ can be paired in the usual
way with differential forms defining smooth valuations. Thus
$\mathscr{E}_{k},\mathscr{F}_{k}\subset\Val_{k}^{-\infty}$ and (10) shows that
$\Val^{\infty}_{k}\subset\mathscr{E}_{k},\mathscr{F}_{k}$ compatible with the
former inclusions.

Let $\tau\in Z_{k}$ and write $\tau=\sum_{I}dx_{I}\wedge\tau_{I}$ with
$\tau_{I}\in\Omega^{n-k}_{L^{2}}(S^{n-1})$, where $I$ runs over all multi-
indices with $|I|=k$. Since $\tau\in Z_{k}$ is an exact generalized form, each
$\tau_{I}$ satisfies the assumption of Lemma 4.8. Thus we can find
$\omega=\sum_{I}dx_{I}\wedge\omega_{I}\in\Omega_{H^{1}}^{k,n-1-k}(S\RR^{n})^{tr}$
such that $d\omega=\tau$ and $\|\omega\|_{1}\leq C\|\tau\|_{0}$. This defines
$\mathscr{F}_{k}\hookrightarrow\mathscr{E}_{k}$. Since
$\Omega_{H^{1}}^{k,n-1-k}(S\RR^{n})^{tr}\hookrightarrow\Omega_{L^{2}}^{k,n-1-k}(S\RR^{n})^{tr}$
is compact by the Rellich–Kondrachov theorem (see, e.g.,
[LawsonMichelson:Spin, Theorem 2.6]), it follows that the inclusion
$\mathscr{F}_{k}\hookrightarrow\mathscr{E}_{k}$ is compact.

######  Remark 4.12.

The space $\Val^{-\infty}_{k}$ is natural in the sense that it is independent
of the choice of a euclidean inner product and an orientation on $\RR^{n}$.
The same is true for the subspaces
$\mathscr{E}_{k},\mathscr{F}_{k}\subset\Val_{k}^{-\infty}$.

The following theorem clarifies the connection between the two Hilbert space
structures.

######  Theorem 4.13 (Poincaré duality).

The Poincaré pairing defined via convolution
$\Val_{k}^{\infty}\times\Val_{n-k}^{\infty}\to\CC$, extends to a continuous
perfect pairing $\mathscr{E}_{k}\times\mathscr{F}_{n-k}\to\CC$. In particular,
the induced map
$\operatorname{Pd}_{k}\colon\mathscr{E}_{k}\to(\mathscr{F}_{n-k})^{*}$ is an
isomorphism of Hilbert spaces and is called _Poincaré duality_.

######  Proof 4.14.

The Hodge star operator
$*\colon\Omega^{k,l}(S\mathbb{R}^{n})^{tr}\to\Omega^{n-k,n-1-l}(S\mathbb{R}^{n})^{tr}$
is an isometry for the $L^{2}$ norms and satisfies $*^{2}=\mathrm{id}$. It is
easy to check that for $1\leq k\leq n-1$ it induces an isometric isomorphism

$*\colon
Z_{n-k}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}Y_{k}^{\perp}\cap
X_{k}$

that extends to the $L^{2}$ completions. The statement of the theorem now
follows easily from (18) and (10).

######  Lemma 4.15.

If $\phi\in\Val^{\infty}_{n-k}$, then convolution with $\phi$ extends to
continuous maps $\mathscr{E}_{l}\to\mathscr{E}_{l-k}$ and
$\mathscr{F}_{l}\to\mathscr{F}_{l-k}$ for all $0\leq k\leq l\leq n$.

######  Proof 4.16.

The cases $k=0$ and $l=n$ are obvious and the case $k=l$ follows from Theorem
4.13. If $0<k<l<n$, we have by Theorem 2.7 well-defined maps $X_{l}/Y_{l}\to
X_{l-k}/Y_{l-k}$ and $Z_{l}\to Z_{l-k}$ corresponding to the convolution with
$\phi$ that are clearly continuous for the $L^{2}$ norm.

###  4.3. Our elliptic operator

Let $\mathbf{C}=(C_{1},\ldots,C_{n-2k})$ be a tuple of convex bodies from
$\mathcal{K}^{\infty}_{+}$, where $0\leq k\leq\frac{n}{2}$. Recall that the
elements of $\Omega^{i,j}(S\RR^{n})^{tr}$ can be identified with the sections
of the bundle $E_{i,j}$ over $S^{n-1}$. We define the differential operators

$\displaystyle P_{0}$
$\displaystyle:=d\colon\Gamma(E_{n-k,k})\to\Gamma(E_{n-k,k+1}),$
$\displaystyle P_{1}$
$\displaystyle:=\alpha\colon\Gamma(E_{n-k,k})\to\Gamma(E_{n-k+1,k}),$
$\displaystyle P_{2}$
$\displaystyle:=d\alpha\colon\Gamma(E_{n-k,k})\to\Gamma(E_{n-k+1,k+1}),$
$\displaystyle P_{3}$
$\displaystyle:=*_{1}^{-1}(\Omega_{\mathbf{C}}\wedge*_{1}\Cdot)\colon\Gamma(E_{n-k,k})\to\Gamma(E_{k,n-k}),$

where $\Omega_{\mathbf{C}}=\Omega_{C_{1}}\wedge\cdots\wedge\Omega_{C_{n-2k}}$.
Note that only $P_{0}$ is a differential operator of positive order . The
operators $P_{1},P_{2},P_{3}$ have order $0$, i.e., are bundle morphisms. Note
that $P_{3}=\mathrm{id}$ if $k=\frac{n}{2}$.

######  Definition 4.17.

Let $P_{\mathbf{C}}\colon\Gamma(E_{n-k,k})\to\Gamma(E_{n-k,k})$ be the
differential operator of order two defined by

$P_{\mathbf{C}}=P_{0}^{*}P_{0}+\sum_{i=1}^{3}P_{i}^{*}\Delta P_{i}.$

The adjoints are formed with respect to the canonical hermitian metric on
$E_{n-k,k}$ and the canonical riemannian metric on $S^{n-1}$.

######  Proposition 4.18.

$P_{\mathbf{C}}$ is elliptic and injective for all $0<k<\frac{n}{2}$.

######  Proof 4.19.

The symbols are given for any $\xi\in T^{*}S^{n-1}$ by

$\displaystyle\sigma_{\xi}(P_{0})(\tau)$
$\displaystyle=\mathbf{i}\xi\wedge\tau,$
$\displaystyle\sigma_{\xi}(P_{1})(\tau)$ $\displaystyle=\alpha\wedge\tau,$
$\displaystyle\sigma_{\xi}(P_{2})(\tau)$ $\displaystyle=d\alpha\wedge\tau,$
$\displaystyle\sigma_{\xi}(P_{3})(\tau)$
$\displaystyle=*_{1}^{-1}(\Omega_{\mathbf{C}}\wedge*_{1}\tau),$

and

$\sigma_{\xi}(P_{\mathbf{C}})=\sigma_{\xi}(P_{0})^{*}\sigma_{\xi}(P_{0})+|\xi|^{2}\sum_{i=1}^{3}\sigma_{\xi}(P_{i})^{*}\sigma_{\xi}(P_{i}).$

Thus for $\xi\neq 0$ we have $\sigma_{\xi}(P_{\mathbf{C}})(\tau)=0$ if and
only if $\sigma_{\xi}(P_{i})(\tau)=0$ for each $i$. The second and third
equation, $\alpha\wedge\tau=0$ and $d\alpha\wedge\tau=0$, imply by Proposition
3.3 that $*_{1}\tau$ is a $(k,k)$-form; the fourth,
$*_{1}^{-1}(\Omega_{\mathbf{C}}\wedge*_{1}\tau)=0$, says that $*_{1}\tau$ is
primitive with respect to the positive $(1,1)$-forms
$\Omega_{C_{1}},\ldots,\Omega_{C_{n-2k}}$. Hence Timorin’s theorem (Theorem
3.1) yields

(20) $(-1)^{k}{*_{1}\tau}\wedge\b{*_1\tau}\wedge\Omega_{\mathbf{C}_{\setminus
1}}\geq 0,$

where $\mathbf{C}_{\setminus 1}=(C_{2},\ldots,C_{n-2k})$, with equality if and
only if $*_{1}\tau=0$. But the first equation implies that $*_{1}\tau$ and
$\b{*_1 \tau}$ are both divisible by $\xi$, which forces equality in (20).
Thus $*_{1}\tau=0$, as claimed.

To prove that $P_{\mathbf{C}}$ is injective, first note that for each
$\tau\in\Gamma(E_{n-k,k})$,

$\langle P_{\mathbf{C}}\tau,\tau\rangle_{L^{2}}=\langle
P_{0}\tau,P_{0}\tau\rangle+\sum_{i=1}^{3}\langle\Delta
P_{i}\tau,P_{i}\tau\rangle_{L^{2}}\geq 0$

is a sum of non-negative terms. Therefore, if $P_{\mathbf{C}}\tau=0$, then
$P_{0}\tau=0$ and $\Delta P_{i}\tau=0$ for $i=1,2,3$.

Recall that $P_{1}\tau\in\Gamma(E_{n-k+1,k})$ and
$P_{3}\tau\in\Gamma(E_{k,n-k})$. Since $1\leq k<n-1$, $\Delta$ is injective on
$\Omega^{k}(S^{n-1})$. It follows that $P_{1}\tau=0$. We thus have $d\tau=0$
and $\alpha\wedge\tau=0$. In particular, there is $\phi\in\Val_{n-k}^{\infty}$
such that $\tau=\tau_{\phi}$.

If $k>1$, $\Delta$ is injective on $\Omega^{n-k}(S^{n-1})$, hence
$P_{3}\tau=0$. If $k=1$, we write
$P_{3}\tau=\sum_{i=1}^{n}f_{i}(u)dx_{i}\wedge\vol_{S^{n-1}}$ with smooth
functions $f_{i}\in C^{\infty}(S^{n-1})$, where
$\vol_{S^{n-1}}\in\Omega^{n-1}(S^{n-1})$ is the volume form. Then on the one
hand, $\alpha\wedge P_{3}\tau=0$ implies that $u_{j}f_{i}(u)=u_{i}f_{j}(u)$
for all $1\leq i,j\leq n$, and on the other hand $\Delta P_{3}\tau=0$ implies
that $f_{i}$ is constant for all $i$. This is only possible if all $f_{i}$
vanish, i.e., if $P_{3}\tau=0$. In both cases we thus get
$L_{\mathbf{C}}\phi=0$. By injectivity in the hard Lefschetz theorem (Theorem
3.11), $\phi=0$ and hence $\tau=0$.

###  4.4. Proof of surjectivity in the hard Lefschetz theorem

Our strategy will be to first prove the existence of a solution $\psi$ in
$\mathscr{E}_{n-k}$ to the equation $L_{\mathbf{C}}\psi=\phi$, where
$\phi\in\Val_{k}^{\infty}$ and $0<k<\frac{n}{2}$. Using our elliptic operator
$P_{\mathbf{C}}$ we will then be able to conclude that $\psi$ must be in fact
smooth. The existence of a weak solution $\psi\in\mathscr{E}_{n-k}$ rests on
an $L^{2}$ estimate which is also a consequence of the ellipticity of
$P_{\mathbf{C}}$.

For $0<k<n$ let
$\tau\colon\mathscr{E}_{n-k}\to\Omega_{H^{-2}}^{n-k,k}(S\RR^{n})^{tr}$ denote
the continuous extension of $\lcur 0,\omega\rcur\mapsto D\omega$.

######  Lemma 4.20.

Let $\psi\in\mathscr{E}_{n-k}$ with $0<k<\frac{n}{2}$. Then

$P_{\mathbf{C}}\tau_{\psi}=P_{3}^{*}\Delta\tau_{L_{\mathbf{C}}\psi}.$

######  Proof 4.21.

Note that both sides are well defined forms in $H^{-4}(E_{n-k,k})$ by Lemma
4.1. Suppose first that $\psi\in\Val^{\infty}_{n-k}$. Then
$\tau_{L_{\mathbf{C}}\psi}=P_{3}\tau_{\psi}$, while $P_{i}\tau_{\psi}=0$ for
$i=0,1,2$, which implies the displayed equation. The general case follows by
approximation, using Lemma 4.1 and Lemma 4.15.

######  Proposition 4.22.

For $0\leq k\leq\frac{n}{2}$, there exists a constant $c>0$, depending on $k$
and $\mathbf{C}$, such that

$\|\psi\|_{\mathscr{F}}\leq c\|L_{\mathbf{C}}\psi\|_{\mathscr{F}}$

for all $\psi\in\mathscr{F}_{n-k}$.

######  Proof 4.23.

The statement is trivial for $k=0$ (with
$c=\frac{1}{n!V(C_{1},\ldots,C_{n})}$) and $k=\frac{n}{2}$ (with $c=1$). Let
$0<k<\frac{n}{2}$. Let $\psi\in\Val^{\infty}_{n-k}$. Since $P_{\mathbf{C}}$ is
injective, Theorem 4.2(d) and Lemma 4.1 imply that

$\|\tau_{\psi}\|_{s}\leq
c\|P_{\mathbf{C}}\tau_{\psi}\|_{s-2}=c\|P_{3}^{*}\Delta\tau_{L_{\mathbf{C}}\psi}\|_{s-2}\leq
c\|\Delta\tau_{L_{\mathbf{C}}\psi}\|_{s-2}\leq
c\|\tau_{L_{\mathbf{C}}\psi}\|_{s},$

where in each step $c$ denotes a possibly different constant. Choosing $s=0$
yields the desired inequality in the smooth case. The general case follows by
approximation, using Lemma 4.15.

We are now ready to complete the proof of the hard Lefschetz theorem.

######  Theorem 4.24.

For any $\phi\in\Val_{k}^{\infty},0\leq k\leq\frac{n}{2}$ there is
$\psi\in\Val_{n-k}^{\infty}$ such that $L_{\mathbf{C}}\psi=\phi$.

######  Proof 4.25.

The statement follows from injectivity of $L_{\mathbf{C}}$ (Proposition 3.11)
for $k=0$ and is trivial for $k=\frac{n}{2}$. We may therefore assume that
$0<k<\frac{n}{2}$. By Proposition 4.10 we may consider $\phi$ as an element of
$\mathscr{E}_{k}$. Define the subspace

$H:=\\{L_{\mathbf{C}}\rho\mid\rho\in\Val_{n-k}^{\infty}\\}\subset\mathscr{F}_{k}$

and the following linear functional on $H$:

$\ell(L_{\mathbf{C}}\rho)=\langle\phi,\rho\rangle,$

where on the right-hand side is the (extended) Poincaré pairing on
$\mathscr{E}_{k}\times\mathscr{F}_{n-k}$. Note that $\ell$ is well defined by
the injectivity part of the hard Lefschetz theorem, Proposition 3.11. Using
the $L^{2}$ estimate of Proposition 4.22, we obtain

$|\ell(L_{\mathbf{C}}\rho)|\leq\|\phi\|_{\mathscr{E}}\|\rho\|_{\mathscr{F}}\leq
c\|\phi\|_{\mathscr{E}}\|L_{\mathbf{C}}\rho\|_{\mathscr{F}},$

hence $\ell$ is a bounded linear functional.

By the Hahn–Banach theorem $\ell$ extends to a continuous linear functional
$\tilde{\ell}$ on $\mathscr{F}_{k}$. By Poincaré duality (Theorem 4.13), there
is $\psi\in\mathscr{E}_{n-k}$ such that
$\tilde{\ell}=\langle\psi,\Cdot\rangle$ and hence

$\langle\phi,\rho\rangle=\ell(L_{\mathbf{C}}\rho)=\langle\psi,L_{\mathbf{C}}\rho\rangle=\langle
L_{\mathbf{C}}\psi,\rho\rangle$

for all $\rho\in\Val_{n-k}^{\infty}$. Consequently, we have
$L_{\mathbf{C}}\psi=\phi$ and therefore
$P_{\mathbf{C}}\tau_{\psi}=P_{3}^{*}\Delta\tau_{\phi}$ by Lemma 4.20. Since
the right-hand side is smooth if $\phi$ is smooth and $P_{\mathbf{C}}$ is
elliptic, Theorem 4.2(a) implies that $\tau_{\psi}$ is smooth and hence
$\psi\in\Val_{n-k}^{\infty}$.

######  Proof 4.26 (Proof of Theorem 1.2(a)).

Proposition 3.11 and Theorem 4.24 imply that the map
$L_{\mathbf{C}}\colon\Val_{n-k}^{\infty}\to\Val_{k}^{\infty}$ is a bijection.
Since $L_{\mathbf{C}}$ is continuous, the open mapping theorem for Fréchet
spaces implies that $L_{\mathbf{C}}$ is an isomorphism of topological vector
spaces.

###  4.5. Continuous dependence on the reference bodies

Identifying a convex body $A$ with its support function $h_{A}$, we can view
$\mathcal{K}^{\infty}_{+}(\RR^{n})$ as a subset of $C^{2}(S^{n-1})$. The
topology inherited from this inclusion is called the $C^{2}$ topology on
$\mathcal{K}^{\infty}_{+}(\RR^{n})$. If $X$ and $Y$ are Banach spaces we
denote by $\mathscr{L}(X,Y)$ the Banach space of bounded linear operators from
$X$ to $Y$.

The reason why we work in the following proposition with the $C^{2}$ topology
on $\mathcal{K}^{\infty}_{+}(\RR^{n})$ and not on
$\mathcal{K}^{2}_{+}(\RR^{n})$, the class of convex bodies with support
function in $C^{2}$ and strictly positive radii of curvature, is that we don’t
know whether the Lefschetz map $L_{\mathbf{C}}$ is surjective for reference
bodies belonging merely to $\mathcal{K}^{2}_{+}(\RR^{n})$.

######  Proposition 4.27.

Let $0\leq k\leq\frac{n}{2}$. The isomorphism
$L_{\mathbf{C}}\colon\Val^{\infty}_{n-k}\to\Val^{\infty}_{k}$ extends to an
isomorphism $L_{\mathbf{C}}\colon\mathscr{F}_{n-k}\to\mathscr{F}_{k}$ of
Hilbert spaces. Moreover, the maps
$\mathcal{K}^{\infty}_{+}(\RR^{n})^{n-2k}\to\mathscr{L}(\mathscr{F}_{n-k},\mathscr{F}_{k})$,

$(C_{1},\ldots,C_{n-2k})\mapsto L_{\mathbf{C}},$

and
$\mathcal{K}^{\infty}_{+}(\RR^{n})^{n-2k}\to\mathscr{L}(\mathscr{F}_{k},\mathscr{F}_{n-k})$,

$(C_{1},\ldots,C_{n-2k})\mapsto(L_{\mathbf{C}})^{-1},$

where the domain is equipped with the $C^{2}$ topology, are continuous.

######  Proof 4.28.

The elliptic estimate Proposition 4.22 implies that the inverse

$L^{-1}_{\mathbf{C}}\colon\Val_{k}^{\infty}\to\Val_{n-k}^{\infty}$

extends continuously to $\mathscr{F}_{k}\to\mathscr{F}_{n-k}$. This proves the
first part.

The continuous dependence of $L_{\mathbf{C}}$ on $\mathbf{C}$ is clear from
the definition of the norm on $\mathscr{F}$. The continuity of
$\mathbf{C}\mapsto(L_{\mathbf{C}})^{-1}$ is an immediate consequence of the
elementary fact that inversion is continuous in the operator norm.

Our next goal is to show that the same conclusion holds if $\mathscr{F}$ is
replaced by $\mathscr{E}$. We will show this using Poincaré duality.

######  Lemma 4.29.

Let $C\in\mathcal{K}^{\infty}_{+}(\mathbb{R}^{n})$. Then the following diagram
commutes:

FF$L^{\mathscr{E}}_{C}$$\operatorname{Pd}_{k+1}$$(L^{\mathscr{F}}_{C})^{*}$$\operatorname{Pd}_{k}$

######  Proof 4.30.

The identity $\langle L_{C}\phi,\psi\rangle=\langle\phi,L_{C}\psi\rangle$
clearly holds for smooth valuations and extends by continuity to
$\phi\in\mathscr{E}_{k+1}$ and $\psi\in\mathscr{F}_{n-k}$.

######  Corollary 4.31.

Proposition 4.27 holds also with $\mathscr{F}$ replaced by $\mathscr{E}$.

######  Proof 4.32.

Lemma 4.29 immediately implies $\operatorname{Pd}_{k}\circ
L_{\mathbf{C}}^{\mathscr{E}}=(L_{\mathbf{C}}^{\mathscr{F}})^{*}\circ\operatorname{Pd}_{n-k}$
or, equivalently,
$L_{\mathbf{C}}^{\mathscr{E}}=(\operatorname{Pd}_{k})^{-1}\circ(L_{\mathbf{C}}^{\mathscr{F}})^{*}\circ\operatorname{Pd}_{n-k}$.
Since taking duals and composition are continuous in the operator norm, the
corollary follows from Proposition 4.27.

##  5\. Hodge–Riemann relations

In this section we use the properties of the Lefschetz isomorphism, in
particular its continuous dependence on the reference bodies, to prove the
Hodge–Riemann relations. As mentioned already in the introduction, the basic
idea is to continuously deform a given tuple of reference bodies
$\mathbf{C}=(C_{0},\ldots,C_{n-2k})$ into euclidean balls
$\mathbf{B}=(B^{n},\ldots,B^{n})$ and to argue that positivity of the
Hodge–Riemann form $q_{\mathbf{C}}$ is preserved during this process.

The Hodge–Riemann form $q_{\mathbf{C}}$ extends to a bounded quadratic form on
$\mathscr{P}_{\mathbf{C}}\subset\mathscr{F}_{n-k}$, the closed subspace of
primitive valuations with respect to $\mathbf{C}$. However, $0$ will always
belong to the spectrum of $q_{\mathbf{C}}$ and therefore we cannot directly
conclude that small perturbations of $q_{\mathbf{C}}$ will remain positive.
Luckily we can also work with the Hilbert space $\mathscr{E}_{n-k}$, in which
$q_{\mathbf{C}}$ is strictly positive. The price we have to pay for this,
however, is that our quadratic form becomes unbounded, which adds another
layer of technical difficulties.

In this and the following section, we will slightly deviate from the previous
notation. Since the additional body $C_{0}$ is important in the definition of
the primitive subspaces and hence in the Hodge–Riemann form, we will denote by
$\mathbf{C}=(C_{0},\ldots,C_{n-2k})$ the complete tuple of reference bodies
and write $\mathbf{C}_{\setminus 0}:=(C_{1},\ldots,C_{n-2k})$ if we wish to
exclude $C_{0}$.

###  5.1. Facts from perturbation theory

For the convenience of the reader, we review here some of the notions
appearing in the statement of the Kato–Rellich theorem and refer to the books
[ReedSimon:I, ReedSimon:II, Kato] for more information.

An operator in a Hilbert space $\mathscr{H}$ is a linear map
$T\colon\operatorname{dom}(T)\to\mathscr{H}$ defined on a subspace
$\operatorname{dom}(T)$ of $\mathscr{H}$. If $\operatorname{dom}(T)$, the
domain of $T$, is a dense subspace of $\mathscr{H}$, then $T$ is said to be
densely defined. Note that if $T$ is densely defined and bounded, then $T$ can
be uniquely extended to all of $\mathscr{H}$.

An operator $T$ is called closed if its graph is a closed subset of
$\mathscr{H}\times\mathscr{H}$. If the closure of the graph of $T$ happens to
be a graph of an operator $\b{T}$, then $T$ is called closable. In this case
$\b{T}$ is the smallest closed extension of $T$.

The adjoint $T^{*}$ of a densely defined operator $T$ on $\mathscr{H}$ is
defined as follows. Let $\operatorname{dom}(T^{*})$ denote the set of all
$x\in\mathscr{H}$ for which there is an $y\in\mathscr{H}$ with $\langle
Tz,x\rangle=\langle z,y\rangle$ for all $z\in\operatorname{dom}(T)$. For each
such $x\in\operatorname{dom}(T^{*})$ one defines $T^{*}x=y$. The adjoint need
not to be densely defined and even $\operatorname{dom}(T^{*})=0$ is possible.
A densely defined operator $T$ is called self-adjoint if
$\operatorname{dom}(T)=\operatorname{dom}(T^{*})$ and $T^{*}=T$; $T$ is called
symmetric if $\operatorname{dom}(T)\subset\operatorname{dom}(T^{*})$ and
$T^{*}|_{\operatorname{dom}(T)}=T$. Another way to express this is to say that

$\langle Tx,y\rangle=\langle x,Ty\rangle$

for all $x,y\in\operatorname{dom}(T)$.

A symmetric operator $T$ is said to be bounded below by $M\in\RR$, if $\langle
Tx,x\rangle\geq M\langle x,x\rangle$ holds for all
$x\in\operatorname{dom}(T)$. $T$ is called (strictly) positive if $T$ is
bounded below by a (strictly) positive number. Operators that are bounded
below by some number are also called semibounded. If $T$ is bounded below by
$M$ and closable, then $\b{T}$ is also bounded below by $M$.

A densely defined operator $T$ is called essentially self-adjoint if $\b{T}$
is self-adjoint. Proving that an operator is self-adjoint can be difficult.
Below we will make use of the following criterion.

######  Theorem 5.1 ([ReedSimon:II, Theorem X.26]).

Let $T$ be a strictly positive symmetric operator. Then the following are
equivalent

$T$ is essentially self-adjoint.

$T$ has dense image.

$T$ has only one semibounded self-adjoint extension.

For a closed operator $T$ one defines its spectrum as follows. A complex
number $\lambda$ is said to be in the resolvent set of $T$ if
$\lambda-T\colon\operatorname{dom}(T)\to\mathscr{H}$ is a bijection. The
resolvent set is denoted $\rho(T)$. The complement of the resolvent set is the
spectrum of $T$ and is denoted by $\sigma(T)$.

The following lemma is a simple consequence of the spectral theorem for
unbounded operators (see [ReedSimon:I, Theorem VIII.4] or [Rudin:FA, Theorem
12.32]).

######  Lemma 5.2.

Let $T$ be a self-adjoint operator on $\calH$. Then $T$ is bounded below by
$M$ if and only if $\sigma(T)\subset[M,\infty)$.

A basic problem in perturbation theory is to show that $A+B$ inherits the
properties of $A$ if $B$ is not too large compared to $A$. This idea of a
“small perturbation” is made precise in the following definition. Let $A$ and
$B$ be densely defined operators on a Hilbert space $\mathscr{H}$. The
operator $B$ is said to be $A$-bounded if
$\operatorname{dom}(A)\subset\operatorname{dom}(B)$ and if there exist
$a,b\in\RR$ such

(21) $\|Bx\|\leq a\|Ax\|+b\|x\|$

for all $x\in\operatorname{dom}(A)$.

######  Theorem 5.3 (Kato–Rellich theorem, [ReedSimon:II, Theorem X.12]).

Suppose that $A$ is self-adjoint, $B$ is symmetric, and $B$ is $A$-bounded
with $a<1$. Then $A+B$ is self-adjoint on $\operatorname{dom}(A)$. Further, if
$A$ is bounded below by $M$, then $A+B$ is bounded below by
$M-\max\\{b/(1-a),a|M|+b\\}$, where $a$ and $b$ are given by (21). In
particular, if $A$ is strictly positive and $b=0$, then $A+B$ is strictly
positive.

We are going to apply the Kato–Rellich theorem to very special perturbations.
The most transparent way to present our argument is to formulate a general
statement.

######  Lemma 5.4 (Homotopy lemma).

Let $T\colon\operatorname{dom}(T)\to\mathscr{H}$ be an essentially self-
adjoint, strictly positive operator on a Hilbert space $\mathscr{H}$. Let
$S_{t}\colon\mathscr{H}\to\mathscr{H}$, $t\in[0,1]$, be a family of bounded
operators with the following properties:

$S_{0}=\id$;

$t\mapsto S_{t}$ is continuous in the operator norm;

each $S_{t}$ is invertible;

the operators $T_{t}:=S_{t}\circ T$ are symmetric. Then $T_{t}$ is for every
$t\in[0,1]$ an essentially self-adjoint, strictly positive operator satisfying
$\operatorname{dom}(\b{T}_{t})=\operatorname{dom}(\b{T})$.

######  Proof 5.5.

First of all observe that since each $S_{t}$ is invertible also $t\mapsto
S_{t}^{-1}$ is continuous in the operator norm. Let
$C:=\max_{t\in[0,1]}\|S_{t}^{-1}\|<\infty$. By continuity of $t\mapsto S_{t}$
on the compact interval $[0,1]$, we find $\epsilon>0$ such that

$\|S_{t_{1}}-S_{t_{2}}\|\leq\frac{1}{2C}$

whenever $|t_{1}-t_{2}|<\epsilon$.

Since $\id\oplus S_{t}$ is an invertible operator on
$\mathscr{H}\times\mathscr{H}$, the closures satisfy
$\b{T}_{t}=S_{t}\circ\b{T}$ and
$\operatorname{dom}(\b{T}_{t})=\operatorname{dom}(\b{T})$. Since
$\b{T}_{t_{1}}-\b{T}_{t_{2}}=(S_{t_{1}}-S_{t_{2}})\circ
S_{t_{1}}^{-1}\circ\b{T}_{t_{1}}$, we obtain for $|t_{1}-t_{2}|<\epsilon$ and
$x\in\operatorname{dom}(\b{T})$ that

$\displaystyle\|(\b{T}_{t_{1}}-\b{T}_{t_{2}})x\|$
$\displaystyle\leq\|S_{t_{1}}-S_{t_{2}}\|\cdot\|S_{t_{1}}^{-1}\|\cdot\|\b{T}_{t_{1}}x\|\leq\frac{1}{2}\|\b{T}_{t_{1}}x\|.$

Hence $\b{T}_{t_{2}}$ is a small perturbation of $\b{T}_{t_{1}}$ in the sense
of the Kato–Rellich theorem, with $a=\frac{1}{2}$ and $b=0$. Thus
$\b{T}_{t_{2}}$ is self-adjoint and strictly positive if $\b{T}_{t_{1}}$ has
these properties. Iterating this argument finitely many times we see that each
$\b{T}_{t}$ is self-adjoint and strictly positive since $\b{T}=\b{T}_{0}$ has
these properties.

###  5.2. Primitive valuations

Assume $1\leq k\leq n/2$ and let us fix a tuple
$\mathbf{C}=(C_{0},C_{1},\ldots,C_{n-2k})$ of convex bodies from
$\mathcal{K}_{+}^{\infty}(\RR^{n})$.

######  Definition 5.6.

The subspace of primitive valuations with respect to the reference bodies
$\mathbf{C}$, denoted $\mathscr{P}_{\mathbf{C}}\subset\mathscr{E}_{n-k}$, is
the kernel of $L_{\mathbf{C}}\colon\mathscr{E}_{n-k}\to\mathscr{E}_{k-1}$.
Analogously, the subspaces of primitive valuations in $\mathscr{F}_{n-k}$ and
$\Val^{\infty}_{n-k}$ are defined. We use superscripts
$\mathscr{E},\mathscr{F},\infty$ if the ambient space is not obvious from the
context.

We define a projection onto the primitive subspace that is in general not
orthogonal, but has other remarkable properties. Its existence relies on the
hard Lefschetz theorem for valuations proved in Sections 3 and 4.

######  Definition 5.7.

We define the operator
$\pi_{\mathbf{C}}\colon\mathscr{E}_{n-k}\to\mathscr{E}_{n-k}$ by

$\pi_{\mathbf{C}}=\id-L_{C_{0}}\circ L_{\mathbf{C},C_{0}}^{-1}\circ
L_{\mathbf{C}},$

where $(\mathbf{C},C_{0})$ denotes the $(n-2k+2)$-tuple
$(C_{0},C_{1},\ldots,C_{n-2k},C_{0})$.

For the spaces $\mathscr{F}_{n-k}$ and $\Val_{n-k}^{\infty}$, the operator
$\pi_{\mathbf{C}}$ is defined by the same formula. Note that
$\pi_{\mathbf{C}}$ is in all cases continuous. We collect several elementary,
but important properties.

######  Lemma 5.8.

In each of the spaces $\mathscr{E}_{n-k}$, $\mathscr{F}_{n-k}$, and
$\Val^{\infty}_{n-k}$, the following assertions hold:

The operator $\pi_{\mathbf{C}}$ is a projection with image
$\mathscr{P}_{\mathbf{C}}$. Its kernel is the image of
$L_{C_{0}}\colon\mathscr{E}_{n-k+1}\to\mathscr{E}_{n-k}$.

Let $\mathbf{C}^{\prime}$ be another $(n-2k)$-tuple of reference bodies. The
restriction
$\pi_{\mathbf{C}}\colon\mathscr{P}_{\mathbf{C}^{\prime}}\to\mathscr{P}_{\mathbf{C}}$
is an isomorphism.

######  Proof 5.9.

(a) is a formal consequence of the definition.

For the proof of (b), let $\phi\in\mathscr{P}_{\mathbf{C}^{\prime}}$ with
$\pi_{\mathbf{C}}\phi=0$. Then $\phi=L_{C_{0}}\psi$ for some
$\psi\in\mathscr{E}_{n-k+1}$. Hence
$0=L_{\mathbf{C}^{\prime}}\phi=L_{\mathbf{C}^{\prime},C_{0}}\psi$, and thus
$\psi=0$ by the hard Lefschetz theorem applied to the $(n-2k+2)$-tuple
$(\mathbf{C}^{\prime},C_{0})$. This shows that the restriction
$\pi_{\mathbf{C}}\colon\mathscr{P}_{\mathbf{C}^{\prime}}\to\mathscr{P}_{\mathbf{C}}$
is injective.

To prove surjectivity, let $\psi\in\mathscr{P}_{\mathbf{C}}$. By the hard
Lefschetz theorem there exists $\eta\in\mathscr{E}_{n-k+1}$ satisfying

$L_{\mathbf{C}^{\prime},C_{0}}\eta=L_{\mathbf{C}^{\prime}}\psi.$

Put $\phi=\psi-L_{C_{0}}\eta\in\mathscr{E}_{n-k}$. Note that
$L_{\mathbf{C}^{\prime}}\phi=0$ and hence
$\phi\in\mathscr{P}_{\mathbf{C}^{\prime}}$. Since the image of $L_{C_{0}}$ is
the kernel of $\pi_{\mathbf{C}}$, we obtain
$\pi_{\mathbf{C}}\phi=\pi_{\mathbf{C}}\psi=\psi$, as required.

Let us explicitly state the following consequence of the lemma.

######  Lemma 5.10.

$\mathscr{P}^{\infty}_{\mathbf{C}}$ is a dense subspace of
$\mathscr{P}^{\mathscr{E}}_{\mathbf{C}}$.

######  Proof 5.11.

Observe that by definition each Lefschetz map $L_{C}$ preserves the subspace
of smooth valuations. Since $\Val^{\infty}_{n-k}$ is dense in
$\mathscr{E}_{n-k}$ by the definition of the latter space and the image of
$\pi_{\mathbf{C}}$ is $\mathscr{P}_{\mathbf{C}}^{\mathscr{E}}$, the claim
follows from the fact that $\pi_{\mathbf{C}}$ preserves the subspace of smooth
valuations.

An immediate consequence of Proposition 4.27 and Corollary 4.31 is

######  Proposition 5.12.

The map

$\pi:\mathcal{K}^{\infty}_{+}(\RR^{n})^{n-2k+1}\to\mathscr{L}(\mathscr{E}_{n-k}),\quad\mathbf{C}\mapsto\pi_{\mathbf{C}}\\\
$

is continuous, where the domain is equipped with the $C^{2}$ topology and the
target is equipped with the operator norm. The same conclusion holds if
$\mathscr{E}$ is replaced by $\mathscr{F}$.

###  5.3. Proof of the Hodge–Riemann relations

Our strategy will be to fit the Hodge–Riemann relations for smooth valuations
into a framework where we can apply the Kato–Rellich theorem in the form of
the homotopy lemma (Lemma 5.4).

Let $\mathbf{B}$ denote the tuple of reference bodies $(B^{n},\ldots,B^{n})$,
where $B^{n}$ is the euclidean unit ball. Consider on
$\mathscr{P}_{\mathbf{B}}^{\mathscr{E}}$ the densely defined quadratic forms

$\displaystyle q_{\mathbf{B}}(\phi,\psi)$
$\displaystyle=(-1)^{k}\phi*L_{\mathbf{B}_{\setminus 0}}\bar{\psi},$
$\displaystyle\pi_{\mathbf{C}}^{*}q_{\mathbf{C}}(\phi,\psi)$
$\displaystyle=q_{\mathbf{C}}(\pi_{\mathbf{C}}\phi,\pi_{\mathbf{C}}\psi)=(-1)^{k}\pi_{\mathbf{C}}\phi*(L_{\mathbf{C}_{\setminus
0}}\circ\pi_{\mathbf{C}}\bar{\psi}),$

for $\phi,\psi\in\mathscr{P}_{\mathbf{B}}^{\infty}$.

######  Lemma 5.13.

The Hodge–Riemann relations are satisfied for $\mathbf{C}$ if and only if
$\pi_{\mathbf{C}}^{*}q_{\mathbf{C}}(\phi,\phi)>0$ for all
$\phi\in\mathscr{P}^{\infty}_{\mathbf{B}},\phi\neq 0$.

######  Proof 5.14.

This is clear, since
$\pi_{\mathbf{C}}:\mathscr{P}^{\infty}_{\mathbf{B}}\to\mathscr{P}^{\infty}_{\mathbf{C}}$
is bijective.

By Proposition 4.10 and Theorem 4.13, the forms $q_{\mathbf{B}}$ and
$\pi_{\mathbf{C}}^{*}q_{\mathbf{C}}$ can be extended to continuous maps
$\mathscr{P}_{\mathbf{B}}^{\infty}\times\mathscr{P}_{\mathbf{B}}^{\mathscr{E}}\to\mathbb{C}$.
Since $\mathscr{P}_{\mathbf{B}}^{\mathscr{E}}$ is a closed subspace of the
Hilbert space $\mathscr{E}_{n-k}$, it is a Hilbert space itself. By the Riesz
representation theorem we may thus write

$\displaystyle q_{\mathbf{B}}(\phi,\psi)$ $\displaystyle=\langle
T_{\mathbf{B}}\phi,\psi\rangle_{\mathscr{E}}$
$\displaystyle\pi_{\mathbf{C}}^{*}q_{\mathbf{C}}(\phi,\psi)$
$\displaystyle=\langle T_{\mathbf{C}}\phi,\psi\rangle_{\mathscr{E}}$

with symmetric operators $T_{\mathbf{B}}$ and $T_{\mathbf{C}}$ on
$\mathscr{P}_{\mathbf{B}}^{\mathscr{E}}$ that are densely defined on
$\mathscr{P}_{\mathbf{B}}^{\infty}$.

The next proposition is crucial in our proof.

######  Proposition 5.15.

There exists a bounded invertible operator
$S_{\mathbf{C}}\colon\mathscr{P}_{\mathbf{B}}^{\mathscr{E}}\to\mathscr{P}_{\mathbf{B}}^{\mathscr{E}}$
such that $T_{\mathbf{C}}=S_{\mathbf{C}}\circ T_{\mathbf{B}}$. Explicitly,

(22) $S_{\mathbf{C}}:=\left(\pi_{\mathbf{B}}\circ L_{\mathbf{B}_{\setminus
0}}^{-1}\circ L_{\mathbf{C}_{\setminus
0}}\circ\pi_{\mathbf{C}}\right)^{*}:\mathscr{P}_{\mathbf{B}}^{\mathscr{E}}\to\mathscr{P}_{\mathbf{B}}^{\mathscr{E}}.$

######  Proof 5.16.

We compute for any $\phi,\psi\in\mathscr{P}_{\mathbf{B}}^{\infty}$

$\displaystyle\langle T_{\mathbf{C}}\phi,\psi\rangle_{\mathscr{E}}$
$\displaystyle=\pi_{\mathbf{C}}^{*}q_{\mathbf{C}}(\phi,\psi)$
$\displaystyle=(-1)^{k}\pi_{\mathbf{C}}\phi*L_{\mathbf{C}_{\setminus
0}}\circ\pi_{\mathbf{C}}\bar{\psi}$
$\displaystyle=(-1)^{k}\phi*L_{\mathbf{C}_{\setminus
0}}\circ\pi_{\mathbf{C}}\bar{\psi}$
$\displaystyle=(-1)^{k}\phi*L_{\mathbf{B}_{\setminus 0}}\circ
L_{\mathbf{B}_{\setminus 0}}^{-1}\circ L_{\mathbf{C}_{\setminus
0}}\circ\pi_{\mathbf{C}}\bar{\psi}$
$\displaystyle=(-1)^{k}\phi*L_{\mathbf{B}_{\setminus
0}}\circ\pi_{\mathbf{B}}\circ L_{\mathbf{B}_{\setminus 0}}^{-1}\circ
L_{\mathbf{C}_{\setminus 0}}\circ\pi_{\mathbf{C}}\bar{\psi}$
$\displaystyle=q_{\mathbf{B}}(\phi,\pi_{\mathbf{B}}\circ
L_{\mathbf{B}_{\setminus 0}}^{-1}\circ L_{\mathbf{C}_{\setminus
0}}\circ\pi_{\mathbf{C}}\psi)$
$\displaystyle=q_{\mathbf{B}}(\phi,S_{\mathbf{C}}^{*}\psi)$
$\displaystyle=\langle S_{\mathbf{C}}\circ
T_{\mathbf{B}}\phi,\psi\rangle_{\mathscr{E}}.$

In the third line we use that
$\mathrm{im}(\pi_{\mathbf{C}}-\mathrm{id})\subset\mathrm{im}(L_{C_{0}})$ by
Lemma 5.8(a), and that $L_{C_{0}}\circ L_{\mathbf{C}_{\setminus
0}}\circ\pi_{\mathbf{C}}\bar{\psi}=L_{\mathbf{C}}\pi_{\mathbf{C}}\psi=0$ since
$\pi_{\mathbf{C}}\bar{\psi}$ is primitive with respect to $\mathbf{C}$.
Similarly, in the fifth line we use that
$\mathrm{im}(\pi_{\mathbf{B}}-\mathrm{id})\subset\mathrm{im}(L_{B^{n}})$, and
$L_{B^{n}}\circ L_{\mathbf{B}_{\setminus 0}}\phi=L_{\mathbf{B}}\phi=0$ since
$\phi$ is primitive with respect to $\mathbf{B}$.

######  Proposition 5.17.

The operator $T_{\mathbf{B}}$ on $\mathscr{P}_{\mathbf{B}}$ is strictly
positive and essentially self-adjoint, where $\mathbf{B}=(B^{n},\ldots,B^{n})$
with $B^{n}$ the euclidean unit ball.

The operator $T_{\mathbf{C}}$ on $\mathscr{P}_{\mathbf{B}}$ is strictly
positive and essentially self-adjoint, where
$\mathbf{C}\in\mathcal{K}^{\infty}_{+}(\RR^{n})^{n-2k+1}$ is an arbitrary
tuple of reference bodies.

The Hodge–Riemann relations hold for the reference bodies $\mathbf{C}$.

######  Proof 5.18.

The proof of (a) is based on a computation involving highest weight vectors.
Since this computation is rather lengthy and requires us to introduce further
notation, we postpone it to the final section.

Let us prove (b). For $t\in[0,1]$ and $i=0,\ldots,n-2k$ we define

$C_{i}(t):=(1-t)B+tC_{i}\in\mathcal{K}^{\infty}_{+}(\RR^{n}),\quad\mathbf{C}(t):=(C_{0}(t),\ldots,C_{n-2k}(t)).$

Clearly $\mathbf{C}(0)=\mathbf{B}$ and $\mathbf{C}(1)=\mathbf{C}$. Keeping the
notation of Proposition 5.15, we define $S_{t}:=S_{\mathbf{C}(t)}$, so that
$T_{\mathbf{C}(t)}=S_{t}\circ T_{\mathbf{B}}$. By Proposition 5.15, each
$S_{t}$ is invertible. Moreover, in view of (22) and the continuous dependence
on the reference bodies guaranteed by Proposition 4.27, the map $t\mapsto
S_{t}$ is continuous in the operator norm. Lemma 5.13 shows that
$T_{\mathbf{C}(t)}$ is symmetric. Since $T_{\mathbf{B}}$ is essentially self-
adjoint and strictly positive by (a), the hypothesis of the homotopy lemma is
satisfied. We thus conclude that $T_{\mathbf{C}}$ is strictly positive and
essentially self-adjoint.

Part (c) follows from (b) and Lemma 5.13.

######  Proof 5.19 (Proof of Theorem 1.1).

The assumption in Theorem 1.1 says that

$\phi:=\sum_{i=1}^{N}x_{i}V(A_{1}^{i},\ldots,A_{k}^{i},\Cdot[n-k])\in\mathscr{P}^{\infty}_{\mathbf{C}}.$

We have

(23)
$\begin{split}q_{\mathbf{C}}(\phi,\phi)=(-1)^{k}\frac{(n-k)!^{2}}{n!}\sum_{i,j=1}^{N}x_{i}x_{j}V(A_{1}^{i},\ldots,A_{k}^{i},A_{1}^{j},\ldots,A_{k}^{j},C_{1},\ldots,C_{n-2k}),\end{split}$

by (8) and the definition of $L_{\mathbf{C}}$. The Hodge–Riemann relations
imply $q_{\mathbf{C}}(\phi,\phi)\geq 0$ with equality if and only if $\phi=0$.

##  6\. Spectral properties

In the final section we will investigate the properties of the Hodge–Riemann
sesquilinear form $q_{\mathbf{C}}$ as a quadratic form on the Hilbert spaces
$\mathscr{P}_{\mathbf{C}}^{\mathscr{E}}$ and
$\mathscr{P}_{\mathbf{C}}^{\mathscr{F}}$. We will complete the proof of the
Hodge–Riemann relations by a direct computation in the case where all
reference bodies are euclidean balls, which is the initial point in the
homotopy argument in the proof of Proposition 5.17. We also clarify our choice
of the Hilbert space $\mathscr{E}$ instead of $\mathscr{F}$ in this last
section by showing that the properties of the Hodge–Riemann sesquilinear form
on $\mathscr{E}$ and $\mathscr{F}$ are very distinct. The former is closable
and has a discrete spectrum of eigenvalues with finite multiplicities
accumulating at $+\infty$, while the latter is bounded and has a spectrum of
eigenvalues with finite multiplicities accumulating at $0$. Using the
$\mathscr{E}$ completion, we may relax the smoothness assumption in Theorem
1.1.

###  6.1. Further facts about self-adjoint operators

Our proofs will rely on certain properties of self-adjoint operators that, for
convenience of the reader, we review in this section.

An operator $A$ with the property that $(A-\mu)^{-1}$ is compact for all
$\mu\in\rho(A)$ is said to have compact resolvent. We will need the following
characterization of such operators.

######  Theorem 6.1 ([ReedSimonIV, Theorem XIII.64]).

Let $A$ be a semibounded self-adjoint operator on $\mathscr{H}$. Then the
following are equivalent:

$(A-\mu)^{-1}$ is compact for some $\mu\in\rho(A)$.

$(A-\mu)^{-1}$ is compact for all $\mu\in\rho(A)$.

$\\{x\in\operatorname{dom}(A)\colon\|x\|\leq 1\text{ and }\|Ax\|\leq b\\}$ is
compact for all $b$.

There exists a complete orthonormal basis $(x_{n})_{n\in\NN}$ in
$\operatorname{dom}(A)$ so that $Ax_{n}=\mu_{n}x_{n}$ with
$\mu_{1}\leq\mu_{2}\leq\cdots$ and $\mu_{n}\to\infty$.

The spectrum of $A$ is discrete and consists of eigenvalues with finite
multiplicities.

Next, we need some facts about quadratic forms and refer to [ReedSimon:I,
Section VIII.6] for details. By a quadratic form $q$ on a Hilbert space
$\mathscr{H}$ we will understand a sesquilinear form
$q\colon\operatorname{dom}(q)\times\operatorname{dom}(q)\to\CC$ defined on a
dense subspace $\operatorname{dom}(q)$ of $\mathscr{H}$. If there is a number
$M\in\RR$ such that $q(x,x)\geq M\|x\|^{2}$ holds for all
$x\in\operatorname{dom}(q)$, then $q$ is said to be semibounded. If
$\mathscr{H}$ is complex, this implies that $q$ is symmetric, i.e.,
$q(y,x)=\overline{q(x,y)}$ for all $x,y$. If $M\geq 0$, then $q$ is said to be
positive. If $|q(x,x)|\leq M\|x\|^{2}$ for some $M$ and all
$x\in\operatorname{dom}(q)$, then $q$ is called bounded.

Let $q$ be a quadratic form bounded below by $M$. $q$ is called closed if
$\operatorname{dom}(q)$ is complete under the norm

$\|x\|_{+1}=\sqrt{q(x,x)+(1-M)\|x\|^{2}}.$

We will need the following fact.

######  Lemma 6.2 ([ReedSimon:II, Problem 15, Chapter VIII]).

A semibounded quadratic form $q$ is closed if and only if whenever

$x_{n}\in\operatorname{dom}(q),\quad
x_{n}\stackrel{{\scriptstyle\mathscr{H}}}{{\to}}x,\quad\text{and}\quad
q(x_{n}-x_{m},x_{n}-x_{m})\underset{m,n\to\infty}{\to}0,$

then $x\in\operatorname{dom}(q)$ and $q(x_{n}-x,x_{n}-x)\to 0$. If this is the
case, we have $q(x,x)=\lim_{l\to\infty}q(x_{l},x_{l})$.

The closure of a quadratic form $q$, if it exists, is the smallest closed
quadratic form which extends it. If a closure of $q$ exists, then $q$ is
called closable.

######  Theorem 6.3 (Friedrichs extension theorem [ReedSimon:II, Theorem
X.23]).

Let $A$ be a positive symmetric operator and let $q(x,y)=\langle Ax,y\rangle$
for $x,y\in\operatorname{dom}(A)$. Then $q$ is a closable quadratic form and
its closure $\hat{q}$ is the quadratic form of a unique self-adjoint operator
$\hat{A}$. $\hat{A}$ is a positive extension of $A$ and the lower bound for
the spectrum of $\hat{A}$ is the lower bound of $q$.

###  6.2. Spectral properties of the operator $T_{\mathbf{B}}$

The paper [KotrbatyWannerer:Harmonic] constructs differential forms
$\omega_{r,k,m}\in X_{r}$ such that the induced valuations
$\phi_{r,k,m}\in\Val_{r}^{\infty}(\RR^{n})$ are highest weight vectors for the
natural action of orthogonal group. Moreover, [KotrbatyWannerer:Harmonic,
Theorem 1.4] computes $q_{\mathbf{B}}(\phi_{r,k,m})$. To prove Proposition
5.17(a), and thereby to complete the proof of the Hodge–Riemann relations, it
will be sufficient to compute the $L^{2}$ norm of the differential forms
$\omega_{r,k,m}$. Let $s_{n}$ denote the volume of the $n$-dimensional unit
sphere.

######  Lemma 6.4.

The valuation $\phi_{r,k,m}\in\Val_{r}^{\infty}$ of
[KotrbatyWannerer:Harmonic, Theorem 1.3] satisfies

$\|\phi_{r,k,m}\|_{\mathscr{E}}^{2}\leq\frac{(n+m-r-2)s_{n+2m-3}}{s_{n+m-r-3}^{2}s_{2m-3}}{n-2k\choose
r-k}.$

######  Proof 6.5.

Using the notation introduced on page 560 of [KotrbatyWannerer:Harmonic], but
moving the constant appearing in front of the integral in
[KotrbatyWannerer:Harmonic, Theorem 1.3] into the definition of
$\omega_{r,k,m}$, we have that

$\omega_{r,k,m}=\frac{(\sqrt{-1})^{\lfloor\frac{n}{2}\rfloor}(\sqrt{2})^{m-2}}{s_{n+m-r-3}}\zeta_{\b{1}}^{m-2}(\omega_{r,k}^{(1)}+\omega_{r,k}^{(2)}),$

where, using the multinomial theorem,

$\displaystyle\omega_{r,k}^{(1)}\otimes\Theta_{1}$
$\displaystyle=\zeta_{\b{K}}(d\zeta_{\b{K}})^{[k-1]}(d\zeta_{L})^{[n-r-k]}(dz_{L})^{[r-k]}(\b{dz_K})^{[k]}$
$\displaystyle=\sum_{a=1}^{k}\sum_{\begin{subarray}{c}A\subset L\\\
|A|=r-k\end{subarray}}(\zeta_{\b{a}}\otimes
dz_{\b{a}})(d\zeta_{\b{K_a}})^{[k-1]}(d\zeta_{L\setminus
A})^{[n-r-k]}(dz_{A})^{[r-k]}(\b{dz_K})^{[k]}$
$\displaystyle=:\sum_{a=1}^{k}\sum_{\begin{subarray}{c}A\subset L\\\
|A|=r-k\end{subarray}}\omega^{(1)}_{a,A}\otimes\Theta_{1},$

and

$\displaystyle\omega_{r,k}^{(2)}\otimes\Theta_{1}$
$\displaystyle=\zeta_{L}(d\zeta_{\b{K}})^{[k]}(d\zeta_{L})^{[n-r-k-1]}(dz_{L})^{[r-k]}(\b{dz_K})^{[k]}$
$\displaystyle=\sum_{a\in L}\sum_{\begin{subarray}{c}A\subset L_{a}\\\
|A|=r-k\end{subarray}}(\zeta_{a}\otimes
dz_{a})(d\zeta_{\b{K}})^{[k]}(d\zeta_{L_{a}\setminus
A})^{[n-r-k-1]}(dz_{A})^{[r-k]}(\b{dz_K})^{[k]}$ $\displaystyle=:\sum_{a\in
L}\sum_{\begin{subarray}{c}A\subset L_{a}\\\
|A|=r-k\end{subarray}}\omega^{(2)}_{a,A}\otimes\Theta_{1}.$

Here $L_{a}=L\setminus\\{a\\}$. Since $\omega^{(1)}_{a,A},\omega^{(2)}_{a,A}$
are both multiples of an orthonormal basis vector, it is readily seen that
pointwise

$\langle\zeta_{\b{1}}^{m-2}\omega^{(i)}_{a,A},\zeta_{\b{1}}^{m-2}\omega^{(j)}_{b,B}\rangle=\delta_{i,j}\delta_{a,b}\delta_{A,B}|\zeta_{\b{1}}|^{2m-4}|\zeta_{a}|^{2}.$

Integration in spherical coordinates gives

$\int_{S^{n-1}}|\zeta_{1}|^{2m-4}\sum_{a=1}^{k}|\zeta_{a}|^{2}d\sigma=\frac{(m+k-2)}{2^{m-2}}\frac{s_{n+2m-3}}{s_{2m-3}}$

and similarly

$\int_{S^{n-1}}|\zeta_{1}|^{2m-4}\sum_{a\in
L}|\zeta_{a}|^{2}d\sigma=\frac{(n-2k)}{2^{m-2}}\frac{s_{n+2m-3}}{s_{2m-3}}.$

Since

$(\omega_{r,k}^{(1)}+\omega_{r,k}^{(2)})\otimes\Theta_{1}=\zeta_{J}(d\zeta_{J})^{[n-r-1]}(dz_{J})^{[r-k]}(\b{dz_K})^{[k]}$

we clearly have $i_{E}\omega_{r,k,m}=0$, where $E$ is the Euler vector field.
Altogether,

$\displaystyle\|\omega_{r,k,m}\|^{2}_{L^{2}}$
$\displaystyle\quad=\int_{S^{n-1}}|\omega_{r,k,m}|^{2}-|i_{E}\omega_{r,k,m}|^{2}$
$\displaystyle\quad=\frac{2^{m-2}}{s_{n+m-r-3}^{2}}\int_{S^{n-1}}|\zeta_{1}|^{2m-4}\left[{n-2k\choose
r-k}\sum_{a=1}^{k}|\zeta_{a}|^{2}+{n-2k-1\choose r-k}\sum_{a\in
L}|\zeta_{a}|^{2}\right]$
$\displaystyle\quad=\frac{(n+m-r-2)s_{n+2m-3}}{s_{n+m-r-3}^{2}s_{2m-3}}{n-2k\choose
r-k}.$

Since $\|\phi_{r,k,m}\|_{\mathscr{E}}=\inf_{\eta\in
Y_{r}}\|\omega_{r,k,m}+\eta\|_{L^{2}}\leq\|\omega_{r,k,m}\|_{L^{2}}$, we
obtain the desired inequality.

######  Proof 6.6 (Proof of Proposition 5.17(a)).

The decomposition of the space
$\mathscr{P}_{\mathbf{B}}\subset\mathscr{E}_{n-r}$ as an orthogonal sum of
irreducible representations under the action of the orthogonal group
$\mathrm{SO}(n)$ was obtained in [KotrbatyWannerer:Harmonic], based on
[ABS:Harmonic]:

(24) $\mathscr{P}_{\mathbf{B}}=\b{\bigoplus_\lambda\Gamma_\lambda},$

where $\lambda$ runs over all highest weights of the form

$(m,\underbrace{2,\ldots,2}_{r-1\ \text{times}},0,\ldots,0),\quad\text{
together with }(m,\underbrace{2,\ldots,2}_{r-2\ \text{times}},-2)\text{ if
}n=2r,$

and where $\Gamma_{\lambda}$ is the irreducible representation of
$\mathrm{SO}(n)$ with highest weight $\lambda$. Moreover,
$\Gamma_{\lambda}\subset\Val^{\infty}_{r}$.

Since $T_{\mathbf{B}}$ is $\SO(n)$-equivariant, by Schur’s lemma,
$\Gamma_{\lambda}$ is an eigenspace of $T_{\mathbf{B}}$. Let $c_{\lambda}$
denote the corresponding (necessarily real) eigenvalue. The valuation
$\phi_{n-r,r,m}$ is a highest weight vector in $\mathscr{P}_{\mathbf{B}}$ with
highest weight $\lambda=(m,2,\dots,2,0,\dots,0)$ and

$\displaystyle\langle
T_{\mathbf{B}}\phi_{n-r,r,m},\phi_{n-r,r,m}\rangle=(n-2r)!\frac{(m+r-1)(n+m-r)v_{n+2m-2}}{v_{r+m-2}^{2}s_{2m-3}}$

by [KotrbatyWannerer:Harmonic, Theorem 1.4]. Thus, using the previous lemma,

$\displaystyle\frac{\langle
T_{\mathbf{B}}\phi_{n-r,r,m},\phi_{n-r,r,m}\rangle}{\|\phi_{n-r,r,m}\|_{\mathscr{E}}^{2}}\geq\frac{(n-2r)!(n+m-r)(r+m-1)(r+m-2)}{(n+2m-2)}.$

Since the number on the right-hand side is $\geq 1$, we conclude that
$c_{\lambda}\geq 1$. Similarly one proceeds for the highest weight
$\lambda=(m,2,\dots,2,-2)$.

If $\phi$ is any element of $\mathscr{P}_{\mathbf{B}}$ and we decompose
$\phi=\sum_{\lambda}\phi_{\lambda}$ according to (24), then

$\langle
T_{\mathbf{B}}\phi,\phi\rangle=\sum_{\lambda}c_{\lambda}|\phi_{\lambda}|^{2}_{\mathscr{E}}\geq\|\phi\|^{2}_{\mathscr{E}}.$

Hence $T_{\mathbf{B}}$ is bounded below by $1$ and in particular strictly
positive.

Since each $c_{\lambda}$ is non-zero, the decomposition (24) also shows that
$T_{\mathbf{B}}$ has dense image. Since $T_{\mathbf{B}}$ is symmetric, Theorem
5.1 implies that $T_{\mathbf{B}}$ is essentially self-adjoint.

###  6.3. The Hodge–Riemann form as a quadratic form on a Hilbert space

The theory of quadratic forms on an infinite-dimensional space is much richer
than in the finite-dimensional setting. In this final section we consider the
Hodge–Riemann sesquilinear form $q_{\mathbf{C}}$ from this perspective.

######  Lemma 6.7.

For every tuple $\mathbf{C}$ of reference bodies from
$\mathcal{K}_{+}^{\infty}(\RR^{n})$, the operator $\b{T}_{\mathbf{C}}$ has
compact resolvent. Moreover, $0$ belongs to the resolvent set.

######  Proof 6.8.

That $\b{T}_{\mathbf{B}}$ has compact resolvent follows from the proof of
Proposition 5.17(a) using Theorem 6.16.1. Recall that
$\b{T}_{\mathbf{C}}=S_{\mathbf{C}}\circ\b{T}_{\mathbf{B}}$, where
$S_{\mathbf{C}}$ is an isomorphism. It follows that for every $b$

(25) $\\{\phi\in\operatorname{dom}\b{T}_{\mathbf{C}}\colon\|\phi\|\leq 1\text{
and }\|\b{T}_{\mathbf{C}}\phi\|\leq b\\}$

is contained in

$\\{\phi\in\operatorname{dom}\b{T}_{\mathbf{B}}\colon\|\phi\|\leq 1\text{ and
}\|\b{T}_{\mathbf{B}}\phi\|\leq b\|S_{\mathbf{C}}^{-1}\|\\}.$

Since $\b{T}_{\mathbf{B}}$ has compact resolvent, the latter set is compact by
Theorem 6.1.

Since (25) is closed, it follows that it is also compact. Using again Theorem
6.1 it follows that $\b{T}_{\mathbf{C}}$ has compact resolvent. Since
$T_{\mathbf{C}}$ is strictly positive by Proposition 5.17(b), so is
$\b{T}_{\mathbf{C}}$ and hence $\sigma(\b{T}_{\mathbf{C}})\subset(0,\infty)$
by Lemma 5.2.

######  Theorem 6.9.

Let $\mathbf{C}=(C_{0},C_{1},\ldots,C_{n-2k})$ be a tuple of reference bodies
from $\mathcal{K}_{+}^{\infty}(\RR^{n})$.

The Hodge–Riemann sesquilinear form $q_{\mathbf{C}}$ defined on
$\mathscr{P}_{\mathbf{C}}^{\infty}\subset\mathscr{E}_{n-k}$ is closable. Its
closure $q^{\mathscr{E}}_{\mathbf{C}}$ is the quadratic form of a unique self-
adjoint operator $Q_{\mathbf{C}}^{\mathscr{E}}$. This operator is strictly
positive. Its spectrum is discrete and consists of eigenvalues with finite
multiplicities.

The Hodge–Riemann sesquilinear form $q_{\mathbf{C}}$ defined on
$\mathscr{P}_{\mathbf{C}}^{\infty}\subset\mathscr{F}_{n-k}$ is bounded. Its
closure $q^{\mathscr{F}}_{\mathbf{C}}$ is the quadratic form of a unique
bounded self-adjoint operator $Q_{\mathbf{C}}^{\mathscr{F}}$. This operator is
positive and compact. Consequently, each non-zero element of the spectrum is
an eigenvalue with finite multiplicity and the eigenvalues accumulate only at
$0$.

######  Proof 6.10.

(a) Note that

$q_{\mathbf{C}}(\phi)=\langle\pi_{\mathbf{C}}^{-*}\circ
T_{\mathbf{C}}\circ\pi_{\mathbf{C}}^{-1}\phi,\phi\rangle_{\mathscr{E}},\quad\phi\in\mathscr{P}_{\mathbf{C}}^{\infty}.$

That $q_{\mathbf{C}}$ is closable follows from the Hodge–Riemann relations and
Friedrichs extension theorem 6.3. Its closure corresponds to a unique self-
adjoint operator $Q_{\mathbf{C}}^{\mathscr{E}}$ and this operator extends
$\pi_{\mathbf{C}}^{-*}\circ T_{\mathbf{C}}\circ\pi_{\mathbf{C}}^{-1}$. But
$\pi_{\mathbf{C}}^{-*}\circ\b{T}_{\mathbf{C}}\circ\pi_{\mathbf{C}}^{-1}$ is
another positive self-adjoint extension. By Theorem 5.1, these extensions must
coincide:
$Q_{\mathbf{C}}^{\mathscr{E}}=\pi_{\mathbf{C}}^{-*}\circ\b{T}_{\mathbf{C}}\circ\pi_{\mathbf{C}}^{-1}$.
By Lemma 6.7, $\b{T}_{\mathbf{C}}$ has compact resolvent. In particular, since
zero belongs to the resolvent set of $\b{T}_{\mathbf{C}}$ by Lemma 6.7,
$\b{T}_{\mathbf{C}}^{-1}$ is compact. It follows that
$(Q_{\mathbf{C}}^{\mathscr{E}})^{-1}$ is compact and Theorem 6.1 shows that
$Q_{\mathbf{C}}^{\mathscr{E}}$ has compact resolvent. This implies the
statement about the spectrum of $Q_{\mathbf{C}}^{\mathscr{E}}$.

(b) Recall from Proposition 4.10 that the inclusion
$\iota_{k}\colon\mathscr{F}_{k}\hookrightarrow\mathscr{E}_{k}$ is compact. Let
$r_{n-k}:\mathscr{F}_{n-k}^{*}\to\mathscr{F}_{n-k}$ be the antilinear map
induced by the Riesz representation theorem, i.e.,
$\langle\phi,r_{n-k}\tau\rangle_{\mathscr{F}}=\langle\phi,\tau\rangle,\tau\in\mathscr{F}_{n-k}^{*},\phi\in\mathscr{F}_{n-k}$.

For $\phi,\psi\in\mathscr{P}^{\infty}_{\mathbf{C}}\subset\mathscr{F}_{n-k}$ we
have

$\displaystyle q_{\mathbf{C}}(\phi,\psi)$
$\displaystyle=(-1)^{k}\phi*L^{\mathscr{F}}_{\mathbf{C}_{\setminus
0}}\b{\psi}$
$\displaystyle=(-1)^{k}\langle\phi,\operatorname{Pd}_{k}\circ\iota_{k}\circ
L_{\mathbf{C}_{\setminus 0}}^{\mathscr{F}}\b{\psi}\rangle$
$\displaystyle=(-1)^{k}\langle\phi,r_{n-k}\circ\operatorname{Pd}_{k}\circ\iota_{k}\circ
L_{\mathbf{C}_{\setminus 0}}^{\mathscr{F}}\b{\psi}\rangle_{\mathscr{F}}$
$\displaystyle=\langle\phi,Q_{\mathbf{C}}^{\mathscr{F}}\psi\rangle_{\mathscr{F}},$

where

$Q_{\mathbf{C}}^{\mathscr{F}}\psi=r_{n-k}\circ\operatorname{Pd}_{k}\circ\iota_{k}\circ
L_{\mathbf{C}_{\setminus 0}}^{\mathscr{F}}\b{\psi}.$

Since the composition of a compact operator with a bounded operator yields a
compact operator, we conclude that $Q_{\mathbf{C}}^{\mathscr{F}}$ is bounded,
self-adjoint, and compact. The Hodge–Riemann relations for smooth valuations
imply that $Q_{\mathbf{C}}^{\mathscr{F}}$ is positive. The statement about the
spectrum is a well-known property of compact operators, see, e.g.,
[ReedSimon:I, Theorem VI.15].

We say that a convex body $K$ in $\RR^{n}$ belongs to the class
$\mathcal{K}^{1,1}(\RR^{n})$ if its support function is differentiable on
$\RR^{n}\setminus\\{0\\}$ and all partial derivatives
$\partial_{i}h_{K}\colon\RR^{n}\setminus\\{0\\}\to\RR$ are locally Lipschitz.
Recall that a function $f\colon A\to\RR$, where $A\subset\RR^{n}$, is called
locally Lipschitz if its restriction to compact subsets of $A$ is Lipschitz.
Moreover, if $U$ is an open subset of $\RR^{n}$, then $f$ is locally Lipschitz
in $U$ if and only if $f\in W^{1,\infty}_{\mathrm{loc}}(U)$, see
[EvansGariepy:Measure, Section 4.2.3, Theorem 5]. Recall also that a function
$f$ belongs to the Sobolev space $W^{1,\infty}_{\mathrm{loc}}(U)$ if and only
if $f\in W^{1,\infty}(V)$ for each compactly contained open subset $V\subset
U$.

######  Lemma 6.11.

Let $A\in\mathcal{K}^{1,1}(\RR^{n})$. Then there exists a sequence $(A_{l})$
of convex bodies from $\mathcal{K}_{+}^{\infty}(\RR^{n})$ such that

$h_{A_{l}}\to h_{A}$ in $C^{1}(\RR^{n}\setminus\\{0\\})$ as $l\to\infty$, and

the partial derivatives

$\left.\frac{\partial^{2}h_{A_{l}}}{\partial x_{i}\partial
x_{j}}\right|_{S^{n-1}}$

converge in $L^{\infty}(S^{n-1})$. Moreover, the limit in (b) does not depend
on the approximating sequence $(A_{l})$ and is denoted by
$\left.\frac{\partial^{2}h_{A}}{\partial x_{i}\partial
x_{j}}\right|_{S^{n-1}}.$

######  Proof 6.12.

The standard mollification procedure, see [Schneider:BM, Theorem 3.4.1] and
the comments after this theorem concerning
$\mathcal{K}_{+}^{\infty}(\RR^{n})$, yields a sequence $(A_{l})$ of bodies
from $\mathcal{K}_{+}^{\infty}(\RR^{n})$ satisfying (a). For
$V=\\{x\in\RR^{n}\colon\frac{1}{2}<|x|\leq 2\\}$ we have $\partial_{i}h_{A}\in
W^{1,\infty}(V)$ and $\partial_{i}h_{A_{l}}\to\partial_{i}h_{A}$ in
$W^{1,\infty}(V)$. Since the second partial derivatives are homogeneous
functions, it follows that

$\left.\frac{\partial^{2}h_{A_{l}}}{\partial x_{i}\partial
x_{j}}\right|_{S^{n-1}}$

converges in $L^{\infty}(S^{n-1})$ and that the limit does not depend on the
sequence $(A_{l})$.

In view of the definition (14) and the preceding lemma we may define a
differential form with $L^{\infty}$ coefficients
$\omega_{\mathbf{A}}\in\Omega_{L^{\infty}}^{n-k,k-1}(S\RR^{n})^{tr}=L^{\infty}(E_{n-k,k-1})$,

$\omega_{\mathbf{A}}:=\frac{(n-k)!}{n!}\cdot*_{1}^{-1}(*_{1}\omega_{A_{1}}\wedge\Omega_{A_{2}}\wedge\cdots\wedge\Omega_{A_{k}}),$

for each tuple $\mathbf{A}=(A_{1},\ldots,A_{k}),1\leq k\leq n$ of convex
bodies from $\mathcal{K}^{1,1}(\RR^{n})$. Let
$\phi_{\mathbf{A}}\in\mathscr{E}_{n-k}$ denote the corresponding valuation.

In the following lemma we use that generalized valuations can be evaluated on
convex bodies belonging to the class $\mathcal{K}_{+}^{\infty}(\RR^{n})$ and
that $\mathscr{E}\subset\Val^{-\infty}$ by Proposition 4.10.

######  Lemma 6.13.

For each tuple $\mathbf{A}=(A_{1},\ldots,A_{k})$ of convex bodies from
$\mathcal{K}^{1,1}(\RR^{n})$, the valuation
$\phi_{\mathbf{A}}\in\mathscr{E}_{n-k}$ is uniquely determined by the property

$\phi_{\mathbf{A}}(K)=V(A_{1},\dots,A_{k},K[n-k])$

for all $K\in\mathcal{K}_{+}^{\infty}(\RR^{n})$. Moreover, if $1\leq
k\leq\frac{n}{2}$ and $\mathbf{C}=(C_{0},\ldots,C_{n-2k})$ is a tuple of
convex bodies from $\mathcal{K}^{\infty}_{+}(\RR^{n})$, then

$L_{\mathbf{C}}\phi_{\mathbf{A}}(K)=\frac{(n-k)!}{(k-1)!}V(A_{1},\dots,A_{k},C_{0},\ldots,C_{n-2k},K[k-1])$

for all $K\in\mathcal{K}_{+}^{\infty}(\RR^{n})$.

######  Proof 6.14.

Choose approximating sequences $(A_{j,l})$ as in Lemma 6.11 and let
$\mathbf{A}_{l}=(A_{1,l},\ldots,A_{k,l})$. Then
$\omega_{\mathbf{A}_{l}}\to\omega_{\mathbf{A}}$ in $L^{\infty}$ as
$l\to\infty$ and by Lemma 3.5 and Theorem 2.7 one has
$\phi_{\mathbf{A}_{l}}=V(A_{1,l},\dots,A_{k,l},\Cdot[n-k])$. Therefore (12)
and Corollary 3.7 imply

$\displaystyle\phi_{\mathbf{A}}(K)$
$\displaystyle=\frac{(-1)^{\binom{n}{2}}}{(n-k)!}\int_{S^{n-1}}*_{1}\omega_{\bf
A}\wedge(\Omega_{K})^{n-k}$
$\displaystyle=\lim_{l\to\infty}\frac{(-1)^{\binom{n}{2}}}{(n-k)!}\int_{S^{n-1}}*_{1}\omega_{\mathbf{A}_{l}}\wedge(\Omega_{K})^{n-k}$
$\displaystyle=\lim_{l\to\infty}V(A_{1,l},\dots,A_{k,l},K[n-k])$
$\displaystyle=V(A_{1},\dots,A_{k},K[n-k]).$

The same argument shows the statement concerning
$L_{\mathbf{C}}\phi_{\mathbf{A}}(K)$.

######  Proposition 6.15.

Let $0\leq k\leq\frac{n}{2}$, $N\in\NN$, $x_{i}\in\RR$, and
$A_{j}^{i}\in\mathcal{K}^{1,1}(\RR^{n})$ and
$C_{i}\in\mathcal{K}^{\infty}_{+}(\RR^{n})$. If

$\phi=\sum_{i=1}^{N}x_{i}V(A_{1}^{i},\dots,A_{k}^{i},\Cdot[n-k])\in\mathscr{E}_{n-k}$

is $\mathbf{C}$-primitive, then $\phi$ lies in the domain of
$q_{\mathbf{C}}^{\mathscr{E}}$ and

(26)
$q_{\mathbf{C}}^{\mathscr{E}}(\phi,\phi)=(-1)^{k}\frac{(n-k)!^{2}}{n!}\sum_{i,j=1}^{N}x_{i}x_{j}V(A_{1}^{i},\dots,A_{k}^{i},A_{1}^{j},\dots,A_{k}^{j},C_{1},\dots,C_{n-2k}).$

######  Proof 6.16.

Choose convex bodies $A^{i}_{j,l}\in\mathcal{K}^{\infty}_{+}(\RR^{n})$ as in
Lemma 6.11. Put

$\psi_{l}=\sum_{i=1}^{N}x_{i}V(A_{1,l}^{i},\dots,A_{k,l}^{i},\Cdot[n-k])\in\Val_{n-k}^{\infty}$

and $\phi_{l}:=\pi_{\mathbf{C}}\psi_{l}$. Then the proof of Lemma 6.13 shows
$\psi_{l}\to\phi$ in $\mathscr{E}_{n-k}$ and hence $\phi_{l}\to\phi$ in
$\mathscr{E}_{n-k}$. Observe that

$\displaystyle q_{\mathbf{C}}(\phi_{l}-\phi_{m},\phi_{l}-\phi_{m})$
$\displaystyle=(-1)^{k}L_{\mathbf{C}_{\setminus
0}}(\phi_{l}-\phi_{m})*\pi_{\mathbf{C}}(\psi_{l}-\psi_{m})$
$\displaystyle=(-1)^{k}\langle L_{\mathbf{C}_{\setminus
0}}^{\mathscr{E}}(\phi_{l}-\phi_{m}),\pi_{\mathbf{C}}^{\mathscr{F}}(\psi_{l}-\psi_{m})\rangle.$

Let $\omega_{l}=\sum_{i=1}^{N}x_{i}\omega_{\mathbf{A}^{i}_{l}}$ and
$\omega=\sum_{i=1}^{N}x_{i}\omega_{\mathbf{A}^{i}}$ be differential forms
representing $\psi_{l}$ and $\phi$. Since
$*_{1}D\omega_{\mathbf{A}}=\frac{(n-k)!}{n!}\cdot\Omega_{A_{1}}\wedge\cdots\wedge\Omega_{A_{n-k}}$,
it follows that $D\omega_{l}-D\omega_{m}\to 0$ in $L^{\infty}$, hence
$\psi_{l}-\psi_{m}\to 0$ in $\mathscr{F}_{n-k}$. We conclude that

$q_{\mathbf{C}}(\phi_{l}-\phi_{m},\phi_{l}-\phi_{m})\to 0\quad\text{as
}l,m\to\infty.$

By Lemma 6.2 and (23), $\phi$ lies in the domain of
$q^{\mathscr{E}}_{\mathbf{C}}$ and

$\displaystyle q^{\mathscr{E}}_{\mathbf{C}}(\phi,\phi)$
$\displaystyle=\lim_{l\to\infty}q_{\mathbf{C}}(\phi_{l},\phi_{l})$
$\displaystyle=(-1)^{k}\frac{(n-k)!^{2}}{n!}\sum_{i,j=1}^{N}x_{i}x_{j}V(A_{1}^{i},\dots,A_{k}^{i},A_{1}^{j},\dots,A_{k}^{j},C_{1},\dots,C_{n-2k}).$

Proposition 6.15 and Theorem 6.9(a) allow us to weaken the hypothesis in
Theorem 1.1 as follows.

######  Corollary 6.17.

The conclusion of Theorem 1.1 holds also if the convex bodies $A_{j}^{i}$
belong merely to $\mathcal{K}^{1,1}(\RR^{n})$.

######  Proof 6.18.

Let the hypothesis of Theorem 1.1 be satisfied for convex bodies $A_{j}^{i}$
from $\mathcal{K}^{1,1}(\RR^{n})$. By Lemma 6.13, the valuation

$\phi=\sum_{i=1}^{N}x_{i}V(A_{1}^{i},\dots,A_{k}^{i},\cdot[n-k])$

is primitive as an element of $\mathscr{E}_{n-k}$ and thus belongs by
Proposition 6.15 to the domain of $q^{\mathscr{E}}_{\mathbf{C}}$. Since
$q^{\mathscr{E}}_{\mathbf{C}}$ is strictly positive by Theorem 6.9(a),
Proposition 6.15 yields

$(-1)^{k}\sum_{i,j=1}^{N}x_{i}x_{j}V(A_{1}^{i},\dots,A_{k}^{i},A_{1}^{j},\dots,A_{k}^{j},C_{1},\dots,C_{n-2k})\geq
0$

with equality if and only if $\phi=0$ as an element of $\mathscr{E}_{n-k}$. By
Lemma 6.13 the latter is equivalent to
$\sum_{i=1}^{N}x_{i}V(A_{1,l}^{i},\dots,A_{k,l}^{i},K[n-k])=0$ for all convex
bodies $K$.

## References