On the extendability by continuity of angular valuations on polytopes
Thomas Wannerer
math.MG 52A20, 52A39 (primary), 52B45, 52B11 (secondary)






A classical theorem of P. McMullen describes all valuations on polytopes that are 
invariant under translations and  weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically  not difficult to check that a valuation is weakly continuous, it is not  clear how to decide whether it admits a continuous extension to convex bodies.
In a special case of McMullen's construction a simple necessary and sufficient condition on the initial data of such an extension is obtained. 
 


Collapse of layer dimerization in the photo-induced hidden state of 1T-TaS_2
    Tobias Ritschel
    December 30, 2023
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§ INTRODUCTION



A  function on the space of  convex bodies in  ^n  
is called a valuation if it satisfies 

    μ(K∪ L) = μ(K)+μ(L)-μ(K∩ L)

 whenever K∪ L is convex. The prime examples of valuations on convex bodies are the intrinsic volumes, but also many other of the most central constructions in Convex Geometry are valuations. Remarkably, in many  cases (<ref>)  together with a short list of inevitable properties determines these constructions uniquely. The seminal result in this direction is Hadwiger's characterization of the intrinsic volumes <cit.>; see, e.g., <cit.> for results in similar spirit. The recent work of Alesker on valuations <cit.> completely transformed this classical line of research by introducing numerous new tools and  opening up  new directions that extend well  beyond Convexity.

The most important examples of valuations on convex bodies are usually invariant under translations and continuous with respect to the Hausdorff metric. The restriction of  such a valuation to polytopes   satisfies (<ref>) for polytopes, is invariant under translations, and is  weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically  not difficult to  check that a valuation is weakly continuous, it is not clear how to decide whether it admits a continuous extension to convex bodies  (see, e.g., <cit.>, <cit.>, and also <cit.> where this is a key issue):


Specify (simple necessary and sufficient) conditions under which a translation-invariant, weakly continuous valuation on polytopes admits a continuous extensions to  convex bodies.



Since polytopes are a dense subset, an extension, if it exists, is unique. Moreover, by a theorem of Groemer <cit.>, a continuous extension is automatically  a valuation. 







 §.§ Statement of the theorem

According to McMullen <cit.>, every translation-invariant, weakly continuous valuation on polytopes can be obtained by a fairly explicit geometric construction. A special case of this construction is the following. 
Let F_k,k+1 be the manifold of pairs (E,ℓ) where E⊂^n is a linear subspace of dimension k and ℓ⊂^n is a 1-dimensional linear subspace orthogonal to E. For every continuous function h F_k,k+1→ define

    μ_h(P)= ∑∫_ℓ⊂ N_FP  h(F, ℓ )   dℓ,
 
where  P⊂^n is a polytope, the sum extends over all k-dimensional faces F of P, F is the translate of the affine hull  of F containing the origin,  N_FP is the normal cone of P at the face F, dℓ is the rotation-invariant probability measure on the space of rays in F^⊥ emanating from the origin, and 
h(F, · ) is considered as an even function on rays in F^⊥. By the easy part of McMullen's theorem <cit.>, μ_h is a translation-invariant, weakly continuous valuation on polytopes.

It may sound surprising, but, as was shown by Alesker <cit.>,  there exist smooth functions h such that μ_h does not admit  a continuous extension to convex bodies. Later, several  authors arrived at the same conclusion in very different ways  <cit.>. In the special case n=3 and k=1 Alesker <cit.> described a subspace of smooth sections  of a certain line bundle over F_1,2 that can be identified with those functions h for which a continuous extensions is possible.  A sufficient condition for all n and k was found by Hinderer, Hug, and Weil <cit.>: if h lies in the image of a certain integral transform, then μ_h extends by continuity. 

The simplest non-trivial case of the construction (<ref>) is when the function h does not depend on ℓ. In this situation, the integrals are proportional to the external angle γ(F,P) of the  polytope P at the face F,  

    μ_π^* f(P) = ∑ f(F) γ(F,P) _k(F),

where  π F_k,k+1→_k(^n), (E,ℓ)↦ E is the projection to the Grassmannian and 
f_k(^n)→.
If f≡ 1 one obtains the intrinsic volumes. Other important examples of such valuations that admit a continuous extension are  the Hermitian intrinsic volumes arising in Hermitian Integral Geometry <cit.> or more generally the constant coefficient valuations (see, e.g., <cit.>). Moreover,  valuations of type (<ref>) are of interest in connection with the Angularity Conjecture <cit.>. 


The problem of characterizing those functions f for which (<ref>) admits a continuous extension  was explicitly stated by Fu   <cit.>.  Although (<ref>) is arguably the simplest possible instance of (<ref>), the results of <cit.> provide almost no information on this problem.   The purpose of this note is to resolve this situation by specifying a simple necessary and sufficient condition under which a continuous extension exists.

Throughout this article we assume that n≥ 3. 

 Let 1≤ k <n-1 and  f_k(^n)→. Then μ_π^* f admits a continuous extension to convex bodies if and only if f is the restriction of a 2-homogeneous polynomial to the image of the Plücker embedding. 
 Consequently, the space of such valuations has dimension

    1/n-k+1nkn+1k+1

and coincides with the space of constant coefficient valuations.


 As was already observed in <cit.> for k=n-1 there is no condition on f except continuity.
Theorem <ref> directly implies the classification of angular curvature measures that has been recently obtain by the author <cit.>.


Let us clarify what we mean by restriction to the image of the Plücker embedding in the statement of theorem. The Plücker embedding  is a map 
    ψ_k(^n) →^k ^n

from the Grassmannian of oriented k-planes in ^n into  the kth exterior power of ^n. If u_1,…, u_k is a positively oriented orthonormal basis of E, then 

    ψ(E)= u_1∧⋯∧ u_k.

We call a function on the oriented Grassmannian even if it is invariant under change of orientation; even  functions on the oriented Grassmannian correspond bijectively to functions on the Grassmannian _k( ^n ) of (unoriented) linear subspaces of ^n. In particular one may restrict even functions on  ^k ^n
to the image of the Plücker embedding to obtain functions on the Grassmannian _k(^n).  






 §.§ Outline of the proof


 We conclude this introduction with a plan of the proof of Theorem <ref>. We denote by _k^+=^+_k(^n) the space of k-homogeneous, continuous valuations on convex bodies in ^n that are invariant under translations and even (i.e., invariant under reflections in the origin).  

 



 A valuation μ∈_k^+ is a called angular if 
 
    μ(P)= ∑_μ(F) γ (F,P) _k(F)

 for every polytope P⊂^n. Here the sum is over the k-dimensional faces F of P,
 _μ_k(^n)→ is the Klain function of μ  (see Section <ref> for its definition),
 F is the translate of the affine hull of F that contains the origin, and γ(F,P) is the external angle of P at F.  The subspace of angular valuations is denoted by A_k⊂_k^+. 




Clearly, if the valuation μ_π^*f extends by continuity to convex bodies, then  μ_π^*f∈ A_k
 and _μ_π^*f = f. The strategy of the proof of Theorem <ref> is to  characterize the Klain functions of the elements of A_k. 
 To this end we pass to the more manageable subspace of smooth valuations introduced by Alesker. This will result in no loss of generality, since, as we will prove  in Section <ref>, smooth valuations are dense in A_k. By a deep theorem of Alesker, the Irreducibility Theorem <cit.>, every smooth valuation can be represented by integration of a differential form over the normal cycle of a convex body (see Section <ref> below for precise statements). This provides us with the computational framework to deduce by an variational argument the  following necessary condition on the Klain function of μ∈ A_n-2. 
 Here and in the following we will not notationally  distinguish between functions on _k(^n) and even functions on unit length  simple k-vectors of ^n. 



  If μ∈ A_n-2, then the function f_2(^n) →, 

    f(E)=_μ(E^⊥),
 satisfies 
 
    (n-1)∫_E_2^n-1 f( 
      1/√(n-1)∑_i=1^n-1ϵ_i u_i ∧ u_n)   dϵ   = 
    ∑_i=1^n-1 f( u_i ∧ u_n)

for every orthonormal basis u_1,…, u_n of ^n, where dϵ denotes the uniform probability measure on E_2^n-1={-1,1}^n-1. 


One may deduce a similar condition for μ∈ A_k. We will however proceed differently and first show that (<ref>) holds if and only if f is the  restriction of a 2-homogeneous polynomial to the image of the Plücker  embedding and then finish the general case by an inductive argument. 







§ PRELIMINARIES





 §.§ Convex geometry


Let us denote by (^n) the space of convex bodies in ^n. Continuity of functions on this space will always be understood with respect to the Hausdorff metric. The convex hull of a subset A⊂^n will be denoted by A. For other basic notions from Convex Geometry not defined here we refer the reader to our standard reference <cit.>.




  §.§.§ Continuous valuations

Alesker's book  <cit.> and Chapter 6 of <cit.> are our references for this material.
Let = (^n) denote the space of translation-invariant continuous valuations on ^n. By McMullen's decomposition theorem, 

    = ⊕_k=0^n_k,
 
where _k denotes the subspace of k-homogeneous valuations, i.e., those valuations μ satisfying μ(tK) = t^kμ(K) for all t>0 and K∈(^n). Moreover, there is the splitting _k= _k^+⊕_k^- into even and odd valuations with respect to reflections in the origin. The space  is equipped with the topology of uniform convergence on compact subsets. In fact,  becomes a Banach space with respect to the norm 

    μ = sup_K ⊂ B |μ(K)|,

where  B is a convex body with nonempty interior.

The affine surface area is a notable exception of a geometrically relevant valuation that does not fit into this framework <cit.>; of a different and more recent vintage are valuations invariant under the indefinite orthogonal groups <cit.>.



Let μ∈_k^+ and let E⊂^n be a k-dimensional linear subspace. By Hadwiger's characterization of volume (see, e.g., <cit.>), the restriction of μ to E is proportional to the k-dimensional volume on E,

    μ|_E = _μ(E) _k.

The map _k^+ → C(_k(^n)) is called the  Klain embedding and _μ is called the Klain function of μ.  By a theorem of Klain <cit.>, _μ=0 implies μ=0.





  §.§.§ Convex cones

If  C⊂^n is a closed convex cone, then the largest linear subspace contained in  C is called the lineality space of C and denoted by L(C). The polar cone to C is 

    C^∘ ={ x∈^n⟨ x,y⟩≤ 0  for  x∈ C}.

The external angle γ(C) of C is defined to be the fraction of L(C)^⊥ taken up by C^∘. More precisely, 

    γ(C) = _n-k-1(S^n-1∩ C^∘)/_n-k-1(S^n-k-1),     where  k= L(C).

It will be important for us that the external angle of C does not depend on the ambient space: If C⊂^m⊂^n, then  γ(C) is the same whether computed in ^m or in ^n.  

The tangent cone of a polytope P at x∈ P is the closed convex cone

    T_x P = cl⋃_t>0 t(P-x)

The lineality space of   T_x P is the linear subspace parallel to the affine hull the unique face F of P 
containing x in its relative interior. Note also that T_xP = T_y P if x,y belong to the relative interior of the same face F;
we denote this common cone by T_FP. The normal cone of P at the face F is 

    N_F P = T_F P ^ ∘.
 
The external angle of a polytope P at a face F is denoted γ(F,P)=γ(T_FP). 


If μ is a valuation on convex bodies in ^n and E⊂^n is a linear subspace,
we denote by μ|_E the restriction of μ to E. 
 Let E⊂^n be a linear subspace. If μ∈ A_k(^n), then μ|_E∈ A_k(E). 


 This follows immediately from _μ|__k(E)
 =_μ|_E and the fact that  the external angle γ(F,P)=γ(T_F P) of P⊂ E ⊂^n is the same whether computed in E or ^n.  







 §.§ Smooth valuations and the normal cycle



The general linear group GL(n) acts on  by (g·μ)(K)= μ(g^-1 K). A valuation 
μ∈ is called smooth, if the map GL(n)→, g↦ g·μ is smooth. 


Let us write S^n=^n× S^n-1 for the sphere bundle of ^n. 
As a set, the normal cycle of a convex body K⊂^n is  the set of outward unit normals to K,

    N(K)={ (x,v)∈ S^n v  is an outward unit normal at  x∈ K}.

 Using the metric projection p_K^n → K (see <cit.>) it is not difficult to see that (x,v)→ x+v is a bi-Lipschitz equivalence between  N(K) and the boundary of the  convex body K+B^n, where B^n is the Euclidean unit ball. We orient normal cycle such that this is map preserves orientations; our convention for the orientation of a domain D⊂^n is that v_1,…,v_n-1∈ T_p ∂ D is positively oriented if ν_p,v_1,…, v_n-1 is, where ν_p is the outward normal at the point p∈∂ D. 
 
 
 In the language of Geometric Measure Theory the preceding paragraph  can be summarized by saying that the normal cycle of a convex body is an integral current. It was proved in <cit.> 
that for a smooth (n-1)-form ω on S^n-1 which is invariant under translations by elements of ^n the function

    K↦∫_N(K)ω
 
defines a smooth valuation. The converse is also true and follows from Alesker's Irreducibility theorem:



 Let 0≤ k≤ n-1. If μ∈_k^∞ then there exists a translation-invariant smooth differential form ω∈Ω^n-1(S^n) of bi-degree (k,n-k-1) such that 
 
    μ(K)= ∫_N(K)ω

 for every convex body K⊂^n. 





 §.§ Grassmannians and exterior powers


Here we  recall the correspondence between simple k-vectors and k-planes in ^n. For more information and proofs see, e.g., <cit.>. 
If w_1,…, w_k is a basis of a linear subspace  E⊂^n, then 

    w_1∧⋯∧ w_k

is a simple k-vector. A different choice of basis yields a proportional k-vector and 

    { v∈^n v∧ w_1∧⋯∧ w_k=0}= E.



A choice of orientation on ^n is equivalent to a choice unit n-vector ω. With respect to this choice the Hodge star operator * ^k^n→^n-k^n is defined by 

    ξ∧ * η= ⟨ξ, η⟩ω,     ξ,η∈^k^n,

where the inner product ⟨ξ,η⟩ is defined by declaring e_i_1∧⋯∧ e_i_k, 1≤ i_1<⋯ < i_k≤ n, for the the standard orthonormal basis e_1,…,e_n of ^n,  to be an orthonormal basis of ^k^n. 
Hence, if the orientation on E^⊥ is chosen such that E⊕ E^⊥ =^n   has orientation ω, then 

    ψ(E^⊥) = * ψ(E),      E∈_k(^n),

where ψ denotes the Plücker embedding (<ref>).

Finally, let us remark that if v_1∧⋯∧ v_k and w_1∧⋯∧ w_k are simple k-vectors then 

    ⟨ v_1∧⋯∧ v_k, w_1∧⋯∧ w_k ⟩ =  (⟨ v_i, w_j⟩ ).

In particular, 

    |v_1∧⋯∧ v_k| = k! _k({0,v_1,…, v_k}).






 §.§ Representation theory of the orthogonal group


We recall from <cit.> the parametrizations of the  irreducible complex representations of the special  orthogonal groups SO(n).
The description of the representations of these groups is sensitive to the parity of the dimension n.
The irreducible representations of SO(2m+1) are parametrized by integer sequences (λ_1, …, λ_m) with 
    λ_1≥…≥λ_m≥ 0;

the irreducible representations of SO(2m) are parametrized by integer sequences (λ_1, …, λ_m) with 
    λ_1≥…≥λ_m  and  λ_m-1≥ |λ_m|.


For example,  the representation ^i V, where V=^n is the standard representation of SO(n), is irreducible for 1≤ i < m  and  corresponds to  

    λ_1=⋯=λ_i=1     and    λ_i+1=⋯ =λ_m=0.

When  n is odd, then also ^m V is irreducible and has highest weight

    λ_1=⋯=λ_m=1;
 
but when n is even  ^m V is not irreducible: the eigenspaces of the Hodge star operator are the irreducible subrepresentations with highest weights 

    λ_1=⋯=λ_m-1=1   and    λ_m=± 1.
 



  §.§.§ Spherical harmonics
 Let  _p^n  denote the subspace of p-homogeneous complex-valued polynomials on ^n  that are harmonic, i.e., satisfy Δ f= 0. The restriction of _p^n to S^n-1 yields precisely the irreducible subrepresentations of C(S^n-1) under the action of the orthogonal group.  Since 
∂^2/∂ x^2 + ∂^2/∂ y^2= 4 ∂^2/∂ z ∂z̅, the polynomial  

    z̅_1^p = (x_1 -ix_2)^p

is in _p^n and a short computation shows that it is in fact a highest weight vector of  weight 
(p,0,…, 0), see, e.g., <cit.>.



  §.§.§ Restriction to the image of the Plücker embedding
 




 The subspace of C(_k(^n)) of restrictions of 2-homogeneous polynomials on ^k ^n  to _k(^n) decomposes under the natural action of 
 SO(n)  precisely into the  irreducible representations with highest weights (2 m_1, 2 m_2,…, 2 m_⌊ n/2⌋) satisfying 
 |m_1|≤ 1 and m_i=0 for i>min(k,n-k). Its dimension is 
 
    1/n-k+1nkn+1k+1.

 

 This is a well-known fact that is proved for example in <cit.> and <cit.>






§ REDUCTION TO SMOOTH VALUATIONS




 A_k⊂_k^+ is closed and invariant under the action of O(n). 


 If (μ_i)_i=1^∞ is a sequence of angular valuations converging uniformly on compact subsets to some valuation μ∈_k^+, then also _μ_i→_μ uniformly. If P is a polytope, then
 
    μ_i(P)= ∑_μ_i(F) γ (F,P) _k(F)

 for every i. Passing to the limit, we see that μ is angular as well.   
It is clear form the definition that for every g∈ O(n) the valuation g·μ is angular if μ is.


Let us recall some terminology from representation theory (see, e.g., <cit.>).
Let K× V→ V be a representation of a group K on an arbitrary vector space V. A vector v∈ V is a called K-finite the span of { k· v k∈ K} is finite-dimensional. For a compact Lie group K, let K be the set of equivalence classes of irreducible finite-dimensional representations of K.  A continuous representation of GL(n) on a Frechét space V is called admissible if for each γ∈O(n) the sum of all irreducible O(n)-invariant subspaces that are in the class γ has finite dimension. The existence of the  Klain embedding _k^+→ C(_k(^n)) implies that the representation of GL(n) on _i^+ is admissible <cit.>.


 A_k∩_k^∞ is a dense subspace of A_k. 


The action of O(n) on A_k is a continuous representation of a compact Lie group on a Banach space. It is well known that in this situation the subspace of O(n)-finite vectors is dense (see, e.g., <cit.>). It is another well-known fact that for every admissible representation of GL(n) on a Frechét space, every O(n)-finite vector is smooth (see, e.g., <cit.>). We conclude that smooth valuations in A_k form a dense subspace. 





§ PROOF OF PROPOSITION <REF>


Throughout  this section let  μ∈_n-2^+(^n) be fixed. Let us also fix an orientation or of ^n and let us  suppose that  μ is given by a translation-invariant smooth differential form ω∈Ω^n-1(S^n) of bi-degree (n-2,1), 
 
    μ(K)=∫_N(K)ω.

 
 
 Consider the simplex S⊂^n spanned by the points p_0=0, p_i=v_i, i=1,…, n-1, p_n=t v_n, where t is a positive number, and v_1,…,v_n is some orthonormal basis of ^n. 
 Let us  make the dependence of the simplex S on the choice of orthonormal basis more explicit by writing 
 S=S(v_1,…, v_n). The first goal of this section is to prove the following lemma.
 Here and in the following for every function f_k(^n)→ and every simple nonzero k-vector w_1∧⋯∧ w_k we will often write 
 f(w_1∧⋯∧  w_k) instead of f(E), E={v∈^n v∧ w_1∧⋯∧  w_k=0}. 
 
    
    lim_t→ 0^+t∫_E_2^n-1μ(S (ϵ_1v_1,…, ϵ_n-1v_n-1, v_n))   dϵ  = 1/4(n-2)!∑_1≤ i< j<n_μ( ⋀_1≤ k≤ n
    k≠ i,j v_k ).

 
 
 
 We will introduce now some notation that will be useful not only in the proof of Lemma <ref>, but also in the proof of Lemma <ref> below.
 Let  F_0,F_1,…, F_n be the facets of S, where the notation is chosen such that p_i∉ F_i. Note that every (n-2)-face of S is of the form 
 
    F_i∩ F_j = { p_k k≠ i,j}.

 The dihedral angle θ_ij∈ (0,π) of this face is given by 
 
    cosθ_ij = ⟨ u_i,u_j⟩,

 where u_i is the facet normal of F_i. Note that 
 
    u_i=-v_i,      i=1,…, n,

 and 
 
    u_0=  1/√(1+(n-1)t^2) (∑_i=1^n-1 tv_i + v_n).

 

 
 
  Let v_1,…, v_n be an orthonormal basis of ^n. Then 
  
    _n-1( { v_1,…, v_n})=  √(n)/(n-1)!.

 

 Note that 
 
    _n-1( { v_1,…, v_n}) = h/n-1_n-2({ v_1,…, v_n-1}),

 where h is the distance from v_n to the affine subspace spanned by v_1,…, v_n-1. 
 Since 
 
    v_n=  1/n-1(v_1 + ⋯ +v_n-1) +  (  v_n - 1/n-1(v_1 + ⋯ +v_n-1))

 where the first vector is in the affine hull of  v_1,…, v_n-1 and the second vector is orthogonal to that affine subspace, we get 
 
    h^2  =  n/n-1.

 Induction on n finishes the proof.


 
 
 We divide the set of all (n-2)-faces of S into four disjoint subsets. 
 
 
  
  * F_i∩ F_n for  i=1,…,n-1. 
 These faces  do not depend on the parameter t and have volume 
 
    _n-2(F_i∩ F_n) =  1/(n-2)!.

 The normal cones   
 
    N_F_i∩ F_nS = { -v_i,-v_n}

 are constant and θ_in=π/2.
 
 
 
  * F_i∩ F_j for distinct i,j=1,… n-1. 
 These faces  depend on t, but their affine hull does not.
Their  volume  is 
 
    _n-2(F_i∩ F_j) =  t/(n-2)!.

The normal cones  
 
    N_F_i∩ F_jS  = { -v_i,-v_j}

are constant and θ_ij=π/2
 
 
 
 
  * F_0∩ F_n.
 This face does not depend on the parameter t, but its normal cone does. By Lemma <ref> we have 
 
    _n-2(F_0∩ F_n) = √(n-1)/(n-2)!.

 The normal cone is 
  
    N_F_0∩ F_nS = { u_0,-v_n}
 and the dihedral angle is
 
    θ_0n(t) = arccos (- 1/√(1+(n-1)t^2)).

 Moreover
 
    S^n-1∩ N_F_0∩ F_n S  = {cosα u_n + sinα w_n  0≤α≤θ_0n},

 where 
 
    w_n=1/√(n-1)∑_i=1^n-1 v_i.

 
 
 
 
  * F_0∩ F_i for i=1,…, n-1. These faces 
 have nonconstant volume, affine hull, normal cone, and dihedral angle.
 It follows from a decomposition similar to (<ref>) that 
 
    _n-2(F_0∩ F_i) = √(1+(n-2)t^2)/(n-2)!.

 The normal cone is 
  
    N_F_0∩ F_iS = { u_0,-v_i}
 and the dihedral angle is
 
    θ_0i(t) = arccos (- t/√(1+(n-1)t^2)).

 Moreover
 
    S^n-1∩ N_F_0∩ F_i S  = {cosα u_i + sinα w_i  0≤α≤θ_0i},

 where 
 
    w_i=1/√(1 +(n-2)t^2 ) (∑_1≤ j≤ n-1
     j≠ i tv_j + v_n) .

 
 
 
 

Observe that 

    μ(S)= ∑_F ∫_F× (N_F S ∩ S^n-1)ω,

where the sum is over all (n-2)-faces of S. Using the preceding division of the (n-2)-faces 
of S, up to an additive constant that does not depend on t we have that    μ(S(v_1,…, v_n))  equals S_1+S_2+S_3, where  
 
    S_1:=∑_1≤ i< j<n (-1)^i+j+n+1 t/(n-2)! or(v_1 ∧⋯∧ v_n)  
      
       ∫_0^π/2⟨ω(-cosα v_i -sinα v_j ), ⋀_1≤ k≤ n
    k≠ i,j v_k ∧ 
      (sinαv̅_i -cosαv̅_j )⟩  d α,   
        S_2:=  1/(n-2)!  or(v_1∧⋯∧ v_n)
      
       ∫_0^θ_0n(t)⟨ω (cosα u_n + sinα w_n), ⋀_1<k<n(v_k-v_1) ∧ (-sinαu̅_n + cosαw̅_n)  ⟩  dα, 
       S_3:= ∑_i=1^n-1(-1)^i/(n-2)! or(v_1∧⋯∧ v_n) 
       ∫_0^θ_0i(t)⟨ω (cosα u_i + sinα w_i(t)  ) ,⋀_1≤ k<n
    k≠ i(v_k-tv_n)∧  (-sinαu̅_i + cosαw̅_i ) ⟩   dα.

Here we always  write  x̅ =(0,x)∈^n⊕^n and x= (x,0)∈^n⊕^n if we form wedge products.  Moreover, we have used (<ref>) to obtain

    | ⋀_1<k<n(v_k-v_1) | = (n-2)! _n-2({ v_1,…, v_n-1}) = (n-2)! _n-2(F_0∩ F_n)

and 

    |⋀_1≤ k<n
    k≠ i(v_k-tv_n)|=(n-2)! _n-2(F_0∩ F_i)

for i=1,…, n-1.
 
Now  we compute the first derivative with respect to t and let t→ 0^+.
Let us make the dependence of S_1(t) on the choice of orthonormal basis explicit by writing  S_1(t)(v_1,…, v_n). We have 

    ∫_E_2^n-1 S'_1(t)( ϵ_1 v_1,…, ϵ_n-1 v_n-1, v_n)  dϵ→∑_1≤ i<j<n  (-1)^i+j+n+1  or( v_1∧⋯∧ v_n ) 
       1/4(n-2)!∫_0^2π⟨ω(cosα v_i + sinα v_j),⋀_1≤ k≤ n
    k≠ i,j v_k∧ (-sinαv̅_i + cosαv̅_j) ⟩
         = 1/4(n-2)!∑_1≤ i< j<n_μ( ⋀_1≤ k≤ n
    k≠ i,j v_k ).



Next we use 

    θ_0n(t)→π,     θ_0n'(t)→ -√(n-1)

as t→ 0^+. Since the integrand does not depend on t, taking derivatives yields 

    S_2'(t)→√(n-1)/(n-2)!  or(v_1∧⋯∧ v_n) ⟨ω(v_n),⋀_1<k<n(v_k-v_1) ∧w̅_n⟩.

Now 

    √(n-1)⋀_1<k<n(v_k-v_1) ∧w̅_n      = (-1)^n ∑_k=1^n-1 v_1 ∧⋯∧v̅_k∧⋯∧ v_n-1+ R,

where R contains all terms that are multiples of  v_i∧v̅_i for some 1≤ i ≤ n-1. After integration over E_2^n-1 the terms coming from R cancel and we get

    ∫_E_2^n-1     S'_2(t)( ϵ_1 v_1,…, ϵ_n-1 v_n-1, v_n)   dϵ→
        (-1)^n/(n-2)!  or(v_1∧⋯∧ v_n)   ⟨ω(v_n), 
    ∑_k=1^n-1 v_1 ∧⋯∧v̅_k∧⋯∧ v_n-1⟩.



Finally, since 

    θ_0i(t)→π/2,     θ_0i'(t)→ 1,      i=1, …, n-1,

we get

    S'_3(t)→∑_i=1^n-1(  (-1)^n+1/(n-2)!or(v_1∧⋯∧ v_n)   ⟨ω( v_n ), v_1 ∧⋯∧v̅_i∧⋯∧ v_n-1⟩
       + (-1)^i/(n-2)!   or(v_1∧⋯∧ v_n)  ∫_0^π/2⟨⟨gradω(-cosα v_i + sinα v_n ), sinα∑_1≤ j≤ n-1
     j≠ i v_j⟩ ,
       ⋀_1≤ k<n
    k≠ iv_k∧  (sinαv̅_i + cosαv̅_n  ⟩  dα
        +   (-1)^i/(n-2)! or(v_1∧⋯∧ v_n)   ∫_0^π/2⟨ω(-cosα v_i + sinα v_n ), ξ⟩    dα),

where 

    ξ:=   . t|_t=0⋀_1≤ k<n
    k≠ i(v_k-tv_n)∧  (-sinαu̅_i + cosαw̅_i(t))  
    
     =   ∑_1≤ k<n
    k≠ i(-1)^k+i+1    v_1∧⋯∧(sinαv̅_i+ cosαv̅_n)_ith position∧⋯∧v_k∧⋯∧ v_n 
       + cosα   v_1∧⋯∧v_i∧⋯∧  
    v_n-1∧v̅_k,

where v_k means that this factor is omitted. 
If we integrate over E_2^n-1, then the first summand will cancel the contributions from S_2(t) and the second summand will vanish. Indeed,  the term

    ∫_0^π/2⟨⟨gradω(-cosα v_i + sinα v_n ),  v_j⟩ ,
    ⋀_1≤ k<n
    k≠ iv_k∧  (sinαv̅_i + cosαv̅_n  ⟩  dα,

where 1≤ j≤ n-1, j≠ i,  will not  change sign if we replace v_j by -v_j. 
Similarly after integration the third summand will vanish since 

    (-1)^k+i+1    v_1∧⋯∧(sinαv̅_i+ cosαv̅_n)_ith position   ∧⋯∧v_k∧⋯∧ v_n
        + cosα   v_1∧⋯∧v_i∧⋯∧  
    v_n-1∧v̅_k,

where 1≤ k<n, k≠ i,
will not  change sign if we replace v_k by -v_k. 


  If μ is angular, then 
  
    lim_t→ 0^+t∫_E_2^n-1μ(S (ϵ_1v_1,…, ϵ_n-1v_n-1, v_n))  dϵ = 1/4(n-2)!∑_1≤ i< j<n-1_μ( ⋀_1≤ k≤ n
    k≠ i,j v_k)
       -1/2πn-1/(n-2)!∫_E_2^n-1_μ ( 
      ∑_i=1^n-1 (-1)^i  ϵ_i ⋀_1≤ k<n
    k≠ iv_k) dϵ +
      1/2π1/(n-2)!∑_i=1^n-1_μ(⋀_1≤ k<n
    k≠ iv_k).





 If μ is angular then, up to an additive  constant which does not depend on t, we have that  μ(S (v_1,…, v_n-1, v_n)) equals  A_1+A_2+A_3, where 
 
    A_1(t)  :=   t/4(n-2)!∑_1≤ i< j<n-1_μ( ⋀_1≤ k≤ n
    k≠ i,j v_k ),
    
      A_2(t)  :=    θ_0n(t)/2π√(n-1)/(n-2)!_μ ( ⋀_ 1<k<n (v_k-v_1)),
    
      A_3(t)  :=   ∑_i=1^n-1θ_0i(t)/2π_n-2(F_0∩ F_i(t))_μ(F_0∩ F_i(t)) 
    
        =    ∑_i=1^n-1θ_0i(t)/2π (-1)^i/(n-2)!or(v_1∧⋯∧ v_n) 
       ∫_0^2π⟨ω (cosα u_i + sinα w_i(t)  ) ,⋀_1≤ k<n
    k≠ i(v_k-tv_n)∧  (-sinαu̅_i + cosαw̅_i  ⟩  dα.

 Taking the first derivative, letting t→ 0^+, and integrating over E_2^n-1,  the term
 A_1(t) yields
 
    1/4(n-2)!∑_1≤ i< j<n-1_μ( ⋀_1≤ k≤ n
    k≠ i,j v_k ).

 For A_2(t) we use (<ref>) and get 
 
    -1/2πn-1/(n-2)!∫_E_2^n-1_μ ( ⋀_ 1<k<n (ϵ_k v_k- ϵ_1 v_1)) dϵ.

For A_3(t) we see as in the proof of Lemma <ref> that after integration over E_2^n-1 the first derivative of 
    _n-2(F_0∩ F_i(t))_μ(F_0∩ F_i(t))
 vanishes. Using (<ref>)  we are left with 

    1/2π1/(n-2)!∑_i=1^n-1_μ(⋀_1≤ k<n
    k≠ iv_k).

To simplify (<ref>) note that 

    ⋀_ 1<k<n ( v_k-  v_1)=  ∑_i=1^n-1 (-1)^i+1⋀_1≤ k<n
    k≠ iv_k
 
and hence (<ref>) equals 
 
    -1/2πn-1/(n-2)!∫_E_2^n-1_μ ( 
      ∑_i=1^n-1 (-1)^i  ϵ_i ⋀_1≤ k<n
    k≠ iv_k)  dϵ.





 Let us first assume that μ∈ A_n-2 is smooth. From Lemma <ref> and Lemma <ref> we conclude that 
 
    (n-1) ∫_E_2^n-1_μ ( 
      1/√(n-1)∑_i=1^n-1 (-1)^i  ϵ_i ⋀_1≤ k<n
    k≠ iu_k)   dϵ = 
      ∑_i=1^n-1_μ(⋀_1≤ k<n
    k≠ iu_k)

  for every orthonormal basis u_1,…, u_n of ^n.
  Applying the Hodge star operator  yields

    * (-1)^i   ⋀_1≤ k<n
    k≠ iu_k = (-1)^n+1 u_i ∧ u_n.

Combining this with (<ref>)  we obtain (<ref>). Since by Lemma <ref> any continuous angular valuation can be approximated by smooth angular valuations, the claim follows. 






§ NECESSARY CONDITION FOR K=N-2.


In the section we will deduce the necessary condition in Theorem <ref> in the special case k=n-2. 


 
 Let f_2 (^n)→ be a continuous function. Then f satisfies the relation (<ref>)
 for every orthonormal basis u_1,…,u_n of ^n if and only if f is the restriction of a 2-homogeneous polynomial on ^2 ^n to the image of the Plücker embedding. 



  

  

Let us first prove that the restriction of every to 2-homogeneous polynomial satisfies (<ref>).
To see this recall that 
if p V → is a homogeneous polynomial of degree k on  a finite-dimensional real vector space V, then there exists a 
symmetric multilinear function  p V^k→, 
 called the polarization of p and denoted by again by p,  which is uniquely determined by the equation
 
    p(a,…, a) = p(a),    for all a∈ V.

For a proof see, e.g., <cit.>. Now if f is the restriction of a 2-homogeneous polynomial p on ^2^n to the image of the Plücker embedding, then expanding the left-hand side of (<ref>) by the bilinearity of the polarization
of p immediately yields the claim. 

To prove the converse, let W⊂ C(_2(^n)) be the subspace of all continuous functions f on _2(^n) with the property that (<ref>) holds for every orthonormal basis 
u_1,…, u_n. 
 We have to show that W is contained in  the space of restrictions of 2-homogeneous polynomials on  ^k ^n to  _2(^n). 
 

Clearly,  the subspace W is closed and  invariant under the action of SO(n). Hence W decomposes into certain irreducible subrepresentations.  The highest weights occurring in the representation C(_2(^n)) are well
known. According to Strichartz <cit.>, if n≥ 4, they all have multiplicity 1 and are of the form 
(2m_1,…, 2m_2,0…, 0) 
with integers satisfying 

    m_1≥   |m_2|    and    if  n>4, m_2≥ 0.

By Lemma <ref>, the proof for n≥ 4 will be finished if we can show that only the highest weights with
 
    m_1≤ 1

 can occur in W. For this it will suffice to show that for all other highest weights the corresponding highest weight vector f_m_1, m_2 is not an element of W. This step will be based on an 
 explicit description of the highest weight vectors given by Strichartz  <cit.> (see also <cit.>).
 
For any subspace E∈_2(^n) choose an orthonormal basis X^1,X^2 of E and consider the corresponding n × 2 matrix 
 
    [ X^1_1 X^2_1;     ⋮     ⋮; X^1_n X^2_n;       ]

of coordinates with respect to the standard basis e_1,…, e_n of ^n.
Let X_j denote the jth row of this matrix. For l≤⌊ n/2 ⌋ let A(l) be the l× 2 matrix whose jth row is X_2j-1 + 
√(-1) X_2j, j=1,…, l. Note that the l× l matrix  A(l)A(l)^t is independent of the choice of the orthonormal basis of E.


A  highest weight vector of the irreducible subrepresentation of C(_2(^n)) with highest weight (2m_1, 2m_2,0,…, 0)  
is given by 

 
  * if m_2≥ 0,  
    f_m_1, m_2= ( A(1)A(1)^t)^m_1-m_2( A(2)A(2)^t)^m_2;
 
 
  * if m_2<0,
 
    f_m_1, m_2= ( A(1)A(1)^t)^m_1-|m_2|( A(2)A(2)^t)^|m_2|.





Let e_1,…, e_n be the standard basis of  ^n and put 

    u_1    = se_1 -c e_3,
    
    u_2    = e_2,
    
    u_i    = e_i+1,      3≤ i≤ n-1,
    
    u_n    = ce_1 +s e_3,

where c=cosϕ and s=sinϕ for brevity.

For this choice of orthonormal basis let us   evaluate  (<ref>)  for a highest weight vector f=f_m_1,m_2. Throughout we assume that m_1≠ 0. 
A straightforward computation shows that 

    ∑_j=1^n-1 f_m_1,m_2( u_j ∧ u_n) = 
                                                             1 + (-1)^m_2 c^2m_1 + (-1)^m_1 s^2m_1,      m_2≠ 0;
    
                                                             1 + (n-3) c^2m_1 + (-1)^m_1 s^2m_1,     m_2= 0,

and 

    (n-1)   ∫_E_2^n-1    f( 
      1/√(n-1)∑_j=1^n-1ϵ_j u_j ∧ u_n )   dϵ =  
       2^m_2/(n-1)^m_1-1∫_E_2^n-1 
      ((n-2)c^2 + 2i ϵ_1ϵ_2 s  )^m_1-m_2 (ϵ_2ϵ_3 cs + i ϵ_1(ϵ_2 s -ϵ_3 c))^m_2 dϵ ,

provided m_2≥0; if m_2<0, then the second factor inside the integral has to be conjugated and m_2 replaced by |m_2|. 
We will distinguish the  cases m_2=0, m_2 even (and different from 0), m_2 odd. 

Let us start with the case m_2=0. Note that by (<ref>) and (<ref>) both sides of (<ref>)
are trigonometric polynomials of degree 2m_1. Comparing the coefficients of the leading term c^2m_1, we get

    (n-2)^m_1/(n-1)^m_1-1 =  (n-2).

This can only hold if m_1=1. 


Next let us treat the case  m_2 even and different from 0. Again by (<ref>) and (<ref>) the leading term on both sides is c^2m_1 and comparing coefficients we get 

    2^|m_2|(n-2)^m_1-|m_2|/(n-1)^m_1-1 =2.

This is equivalent to 

    ( n-2/n-1) ^m_1-1  = ( n-2/2) ^|m_2|-1.

Since the left-hand side is strictly less than 1 and the right-hand side greater equal 1, this is a contradiction. Hence no highest weight vector f=f_m_1,m_2 with 
m_2≠ 0 even can satisfy (<ref>).

Let us finally consider the case m_2 odd. Expanding  the two factors inside the  sum in (<ref>) we get 

    ((n-2)c^2)^m_1-m_2 +m_1-m_21     ((n-2)c^2)^m_1-m_2-1 2i ϵ_1ϵ_2 s  
         + m_1-m_22 ((n-2)c^2)^m_1-m_2-2  (2i  s)^2+ ⋯

and 

    ϵ_2ϵ_3 (cs)^m_2 +     m_21  (cs)^m_2-1   iϵ_1(ϵ_2 s- ϵ_3 c)
       + 
    m_22ϵ_2ϵ_3  (cs)^m_2-2 (iϵ_1(ϵ_2 s- ϵ_3 c))^2 + ⋯

where ⋯ denotes lower order terms.  Observe that 

    (iϵ_1(ϵ_2 s- ϵ_3 c))^2= 2ϵ_2ϵ_3 cs-1.

Integrating over E_2^n-1 we see that the leading term c^2m_1-2 on the left-side of (<ref>) results from the product of the second terms in the above expansions and the product of the first in (<ref>) with the third in (<ref>). Comparing coefficients we get 

    (-1)^(m_2-1)/22^m_2+1(n-2)^m_1-m_2-1/(n-1)^m_1-1((n-2)m_22  + (m_1-m_2)m_2)
    = - m_1.
 
Consequently, (m_2-1)/2 must be odd and in particular m_2≥ 3. 
If n=2k+2 is even, then

    2^m_1 k^m_1-m_2-1((n-2)m_22  + (m_1-m_2)m_2)= m_1 (2k+1)^m_1-1.

We conclude that 2^m_1 divides m_1, a contradiction. If n=2k+1 is odd (k≥ 2), then 

    (2k-1)^a ((2k-1) m_22+  (a+1)m_2 )
      =  m_1 2^a-1 k^m_2+a,

where we put a= m_1-m_2-1. 
Since 2l ≤ k^l for integers k,l≥ 2, we have  

    (2k-1)^a   ( (2k-1)   m_22    +  (a+1)m_2 ) 
       ≤  m_2 (2k-1)^a+1k^m_2-1/4 
    + (2k-1)^a (a+1) m_2 
        < m_2 2^a-1 k^m_2+a +  (2k-1)^a (a+1) m_2
        = m_1 2^a-1 k^m_2+a + (a+1)( (2k-1)^a m_2 - 2^a-1  k^m_2+a) 
       ≤ m_1 2^a-1 k^m_2+a,

which contradicts (<ref>). 
Therefore if m_2 is odd and (<ref>) holds for f_m_1,m_2, then we must have m_1-|m_2|=0. 

Now if m_1=|m_2|≥ 3 is odd, then after integration over E_2^n-1 we see that  the leading terms in (<ref>) and (<ref>)  are (cs)^m_1-1 and c^2m_1-2. Comparing coefficients we get 

    (-1)^(m_1-1)/22^m_1/(n-1)^m_1-1 (m_1-1)   = -1.

For the signs to match we must have m_1=4k+3 with k=0,1,2,…. Hence

    4 ·  2^4k+2 (2k+1) = (n-1)^4k+2

which implies that n-1 must contain the prime factor 2 more then once. But then 
(<ref>) can only hold if k=0 and n-1=4.

Now  for n=5 and m_1=|m_2|=3 one immediately checks that (<ref>) and (<ref>) do coincide. Hence we are going to choose a different orthonormal basis of ^n:

    u_1     = c(ae_1+ be_3)+ s e_n,
    
      u_2     = s(ae_1+be_3)-c e_n,
    
       u_3     = e_2,
    
       u_i     = e_i,       4≤ i≤ n-1, 
    
       u_n     = -be_1+ae_3,

where a=cosψ , b=sinψ for brevity.
A straightforward computation shows that 

    ∑_j=1^n-1 f_m_1,± m_1( u_j ∧ u_n) =c^2m_1+s^2m_1 +(-1)^m_1 (a^2m_1+b^2m_1)

and 

    (n-1) ∫_E_2^n-1     f_m_1,± m_1( 
      1/√(n-1)∑_j=1^n-1ϵ_j u_j ∧ u_n )   dϵ = 
       2^m_1/(n-1)^m_1-1∫_E_2^n-1 
    ( ϵ_1ϵ_2 cs - ϵ_3ϵ_4 ab + i (ϵ_1 c +ϵ_2 s) (ϵ_3 a +ϵ_4 b))^m_1 dϵ.

Comparing the coefficients the leading term  we get a contradiction for m_1>1. We conclude that the validity of (<ref>) for a highest weight vector f_m_1,m_2 implies m_1≤ 1.

It remains to treat the case n=3. Taking orthogonal complements, we have to show that if an even  continuous  function 
f S^2→ satisfies 

    2∫_E_2^2  f(1/√(2)∑_j=1^2 ϵ_j u_j) dϵ =∑_j=1^2 f(u_j)

for every pair of orthonormal vectors u_1,u_2, then f is the restriction of a 2-homogeneous polynomial to S^2. According to (<ref>) the function f=f_m_1= (x_1-ix_2)^2m_1, is contained  in the irreducible subrepresentation with highest weight (2m_1). Now if m_1>0, then choosing as before u_1=se_1-ce_3, u_2=e_2, evaluating  both sides, and comparing the coefficients of the leading term  s^2m_1,  we get

    1/2^m_1-1 =1.

Hence also in this case m_1≤ 1, as desired.




 If  μ∈ A_n-2(^n), then  the Klain function of μ is the restriction of a 2-homogeneous polynomial on ^n-2^n to the image of the Plücker embedding. 


 Proposition <ref>, Proposition <ref>, and (<ref>) imply that if μ∈ A_n-2, then the Klain function of μ is the restriction of a 2-homogeneous polynomial on ^n-2^n to the image of the Plücker embedding.
 






§ PROOF OF THEOREM <REF>


In this final section we provide an inductive argument that allows us to deduce Theorem <ref> from the special case k=n-2 already established in  Corollary <ref>. We will need the following fact. 


 Let 1≤ k <n-1 and f be an even continuous function on _k(^n). Suppose that for every nonzero v∈^n there exists a 
 2-homogeneous polynomial q_v on ^k v^⊥ such that f=q_v on _k(v^⊥)⊂^k v^⊥. Then there exists  
  a globally defined 2-homogeneous polynomial q on ^k ^n such that 
 f=q on _k(^n)⊂^k ^n.




  Let  1≤ k <n-1 and f be an even continuous function on _k(^n). Suppose that for every linear subspace  E⊂^n of dimension k+2 there exists a 
 2-homogeneous polynomial q_E on ^k E such that f=q_E on _k(E)⊂^k E. Then there exists  
  a globally defined 2-homogeneous polynomial q on ^k ^n such that 
 f=q on _k(^n)⊂^k ^n.


 Put m=n-k-2. If m=0, then there is nothing to prove. Assume now that m>0 and that the theorem has been proved for m-1. Then for every nonzero vector v∈^n there exists a 
 2-homogeneous polynomial q_v on ^k v^⊥ such that f=q_v on _k(v^⊥)⊂^k v^⊥. By Proposition <ref> this implies the existence of 
  a globally defined 2-homogeneous polynomial q on ^k ^n such that 
 f=q on _k(^n)⊂^k ^n. 





 
If μ=μ_π^*f  admits a continuous extension to  convex bodies then μ∈ A_k.  
For every linear subspace E⊂^n the restriction μ|_E of μ to E is according to Lemma <ref>  angular. The Klain function of μ|_E is the restriction of the Klain function of μ to _k(E). If the dimension of E is k+2, then f|__k(E)=_μ|_E is by Corollary <ref> the restriction of a 2-homogeneous polynomial on ^k E   to the image of the Plücker embedding. By  Corollary <ref>, f is the restriction of 2-homogeneous polynomial on ^k ^n to the image of the Plücker embedding. 
 
The converse follows from the fact proved in <cit.> that to each 
restriction of a 2-homogeneous polynomial  to the image of the Plücker embedding corresponds a constant coefficient curvature measure. Globalizing this curvature measure we get an element of A_k. That   A_k has the stated dimension is now immediate from  Lemma <ref>. 



abbrv

