On the $m$th-order Affine Pólya-Szegö Principle
Dylan Langharst Michael Roysdon Yiming Zhao

§ ABSTRACT 
An affine Pólya-Szegö principle for a family of affine energies, with equality condition characterization, is demonstrated. In particular, this recovers, as special cases, the L^p affine Pólya-Szegö principles due to Cianchi, Lutwak, Yang and Zhang, and subsequently Haberl, Schuster and Xiao. Various applications of this new Pólya-Szegö principle are shown.



What makes a face looks like a hat: Decoupling low-level and high-level Visual Properties with Image Triplets
    Maytus Piriyajitakonkij1,20000-0002-7610-8953 Sirawaj Itthipuripat30000-0001-9302-0964 Ian Ballard†*,40000-0003-1814-3141 Ioannis Pappas*,50000-0002-0168-7014
    September 9, 2024
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§ INTRODUCTION


The sharp L^p Sobolev inequality states that for p∈ (1,n) and f∈ W^1,p(), we have

    a_n,pf_L^np/n-p()≤∇ f_L^p(),

where ·_L^q() stands for the usual L^q norm of a function. The constant a_n,p is the best possible and can be explicitly computed, see (<ref>) below.  The Sobolev inequality has a wide range of applications across all areas of mathematics, particularly in the study of PDEs. In this sharp form, it can be found in Federer and Fleming <cit.>, Fleming and 
Rishil <cit.>, Maz'ya <cit.> for p=1, and Aubin <cit.> and Talenti <cit.> for p∈ (1,n). The Sobolev inequality is arguably the most fundamental inequality connecting analysis and geometry. Indeed, the geometric core behind (<ref>), for all p∈ (1,n), is the classical isoperimetric inequality:

    nω_n^1/nvolume (E)^n-1/n≤surface area(E),

where ω_n is the volume of the n-dimensional unit ball. In fact, the two inequalities are equivalent; see, e.g., Gardner <cit.>. Moreover, the standard approaches to both inequalities are perfectly parallel. A critical ingredient for the proof of the Sobolev inequality is the Pólya-Szegö principle stating that the L^p Dirichlet energy ∇ f_L^p() is non-increasing if f is replaced by its spherically symmetric arrangement whereas the classical isoperimetric inequality can be shown by establishing the monotonicity of the surface area with respect to Steiner symmetrization.  

By now, there is a vast library of literature on Sobolev inequalities and Pólya-Szegö principles. We refer the readers to the survey <cit.> by Talenti, a few recent contributions <cit.> and the references therein.

By convex body, we mean a compact convex set in ^n with non-empty interior. The projection body Π K of a convex body K is an important and natural (see, Ludwig <cit.>) object in affine geometry. As the name suggests, the projection body encodes information about the area of the image of the orthogonal projection of K onto (n-1)-dimensional subspaces. Crucially, the fact that the projection body operator is (n)-contravariant, or Πϕ K= ϕ^-tΠ K for all ϕ∈(n), makes its volume and the volume of its polar body fundamental affine invariants. Recalling that, when a convex body K contains the origin in its interior, its polar body K^∘ is well-defined, the polar projection body of a convex body K is precisely Π^∘ K :=(Π K)^∘. Then, the celebrated Petty projection inequality states that for every convex body K in , we have, denoting by  the Lebesgue measure on ,

    (Π^∘ K) (K)^n-1≤(Π^∘) ()^n-1,

where  is the centered unit ball in  and equality holds if and only if K is an ellipsoid. By Cauchy's area formula and a simple application of Hölder's inequality, it can be seen that the Petty projection inequality trivially implies the classical isoperimetric inequality. Quoting Gardner and Zhang <cit.>:“the Petty projection inequality is a dramatic improvement upon the classical isoperimetric inequality.” Part of the dramatic improvements of Petty's inequality over its classical counterpart is that (<ref>) is affine invariant, or, more precisely, changing K into ϕ K for any ϕ∈(n) will not change the value of the volume product (Π^∘ K) (K)^n-1. It is worth pointing out that the inequality involving the volume ratio (Π K) (K)^-n+1, n ≥ 3, known as the Petty conjecture, is still open <cit.>. 

In a landmark work <cit.>, Zhang, using Minkowski's existence theorem, formulated and derived from the Petty projection inequality, an affine Sobolev inequality that is stronger than (<ref>): 

    a_n,1f_L^n/n-1()≤(nω_n)^n+1/n/2ω_n-1(∫_(∫_ |(∇ f(x))^t u| dx)^-ndu)^-1/n: = ℰ_1(f),
 
where equality holds when f is the characteristic function of an ellipsoid. Zhang <cit.> originally stated (<ref>) for C^1 functions with compact support. However, the inequality holds in general for functions of bounded variation as shown by Wang <cit.>. We emphasize again that not only is (<ref>) sharp, but the energy functional ℰ_1(f) is affine invariant in the sense that ℰ_1(f)=ℰ_1(f∘ϕ) for ϕ∈(n). Moreover ℰ_1(f)≤∇ f_L^1() and therefore, we may view ℰ_1(f) as the affine analog of the classical Dirichlet energy. 

In fact, Lutwak, Yang, and Zhang <cit.> showed that (<ref>) is only one of a family of sharp L^p affine Sobolev inequality: for p∈ [1,n),

    a_n,pf_L^np/n-p()≤ℰ_p(f),

each of which is stronger than its classical counterpart (<ref>). Here, the precise formulation of ℰ_p(f) can be found in <cit.>, or, by substituting m=1 and Q=[-1/2,1/2] in (<ref>). Curiously, while the geometric core behind the L^p Sobolev inequality for any p∈ [1,n) is the same (the isoperimetric inequality), the geometric core behind (<ref>) is an L^p version of the Petty projection inequality (different for each p) established in Lutwak, Yang, and Zhang <cit.>. An asymmetric (and even stronger) version of (<ref>) is due to Haberl and Schuster <cit.>. The proofs invariably use various elements from the booming L^p Brunn-Minkowski theory initiated by Lutwak <cit.> in the 1990s, including the L^p Minkowski inequality and the L^p Minkowski problem, see, e.g., <cit.>.

As in the Euclidean case, an affine Pólya-Szegö principle was established by Lutwak, Yang, and Zhang <cit.> and Cianchi, Lutwak, Yang, and Zhang <cit.> for the L^p affine energy: for p≥ 1,

    ℰ_p(f)≥ℰ_p(f^⋆),

where f^⋆ is the spherically symmetric rearrangement of f. See Section <ref> for its precise definition. An asymmetric (and even stronger) version of (<ref>) is due to Haberl, Schuster, and Xiao <cit.>. An equality condition characterization for p>1 was found in Nguyen <cit.>.

Recently, the first two named authors and their collaborators <cit.> established a higher-order version of the sharp L^p affine Sobolev inequality (<ref>). In particular, their work encompasses <cit.> as special cases. While not explicitly stated, it will become clear in the current work that each inequality in this family is stronger than its classical counterpart.

One of the goals of the current work is to show the validity of the accompanying higher-order affine Pólya-Szegö principle along with its equality condition characterization. 


A convex body K is uniquely determined by its support function h_K:→ given by h_K(u) = max_x∈ K x^t u. The covariogram function of K is given by 

    g_K(x) = (K∩ (K+x)).

The projection body is closely connected to the covariogram of a convex body, see Matheron <cit.>: for θ∈,

    .d/dr|_r=0^+ g_K(rθ) = - [n-1] (P_θ^⊥K) = -h_Π K(u),

where P_θ^⊥K is the image of the orthogonal projection of K onto the (n-1)-dimensional subspace θ^⊥ consisting of vectors perpendicular to θ. 

Let m∈ℕ. For x∈^nm, we will write x= (x_1,…, x_m) where x_i∈. From time to time, we shall identify ℝ^nm with the space of n by m matrices. Consequently, as an example,  the notation u^tx for u∈ and x∈ℝ^nm is simply the usual matrix multiplication between a 1 by n row vector u^t and an n by m matrix x and the product yields a 1 by m row vector.  


The motivations of <cit.> stem from the  investigation of Schneider <cit.> of the the following natural generalization of the covariogram function: for each m∈ℕ, the mth-covariogram function of a convex body K in ^n, is given by

    g_K,m(x) = (K∩⋂_i=1^m(K+x_i)),
 
for x=(x_1,…, x_m)∈ℝ^nm. This seemingly innocent generalization turns out to lead to unexpected things. We will now describe one conjecture following this. Since the support of the classical covariogram g_K gives an alternative definition for the difference body DK= K+(-K), the support of g_K,m, denoted by D^m(K), naturally generalizes the classical difference body. As was noted by Schneider, the set D^m(K) is a convex body in ^nm, and the volume ratio

    [nm](D^m K)[n](K)^-m

is (n)-invariant. It was shown in <cit.> that (<ref>) is maximized by simplices, which in the case of m=1 recovers the celebrated difference body inequality of Rogers and Shephard <cit.>, a certain reverse form of the celebrated Brunn-Minkowski inequality <cit.>.  From this point of view, 
finding the sharp lower bound of (<ref>) may be viewed as a higher-order analog of the Brunn-Minkowski inequality. 
The conjecture is settled in dimension 2: Schneider showed that it is minimized by origin-symmetric K when n =2 for all m ∈. The case for arbitrary n when m=1 is the Brunn-Minkowski inequality for the difference body. The minimizer of (<ref>) in the case n ≥ 3, m ≥ 2 remains open and is conjectured to be attained by ellipsoids. With this point of view in mind, it seems apt to refer to Schneider's conjecture as the mth-order Brunn-Minkowski conjecture.  As observed in <cit.>, this conjecture is intimately connected to Petty's conjecture mentioned above. With the tools developed in <cit.>, Haddad in <cit.> confirmed a version of the  mth-order Brunn-Minkowski conjecture by replacing the volume with mean width. 
 
It was shown in <cit.> that g_K,m is differentiable in each radial direction at the origin. Inspired by Matheron's formula (<ref>), one naturally obtains, for each m∈ℕ, an mth-order projection body; this corresponds to the case when Q is the orthogonal simplex in the following definition, introduced in <cit.>.
  Let p ≥ 1, m ∈, and fix a convex body Q containing the origin in ^m. Given a convex body K containing the origin in its interior in ^n, we define the (L^p, Q)-projection body of K, K, to be the convex body whose support function is given by 

    h_ K(u)^p = ∫_ h_Q(v^tu)^p dσ_K,p(v)

for u∈.

Here σ_K,p is the L^p surface area measure of K, see Section <ref> for its definition. It is simple to see that  is (n) “contravariant”; that is, ϕ K = ϕ^-t K for each ϕ∈(n), where ϕ^-t(x_1,…, x_m)= (ϕ^-t x_1, …, ϕ^-t x_m). Moreover, when m=1, Q=[-1/2,1/2], it recovers the classical L_p projection body Π_p K and when m=1, Q=[0,1], it recovers the asymmetric L^p projection body Π_p^+K first considered in Lutwak <cit.>. It was shown in <cit.>  that K contains the origin as an interior point. Consequently, its polar body ( K)^∘ will simply be denoted by K. 

In <cit.>, a brand new family of sharp affine isoperimetric inequality (containing the Petty projection inequality as a special case) was shown: for p≥ 1, m∈ℕ, and a convex body K in  containing the origin in its interior,

    ( K) (K)^nm/p-m≤() ()^nm/p-m.

For p=1, equality holds if and only if K is an ellipsoid. For p>1, equality holds if and only if K is an origin-symmetric ellipsoid. Naturally, a new collection of sharp affine Sobolev inequality accompanies <cit.>: for p∈ [1,n), m∈ℕ, a convex body Q in ^m containing the origin, and f∈ W^1,p(^n), we have

    a_n,pf_L^np/n-p()≤ d_n,p(Q)(∫_( ∫_ h_Q((∇ f(z))^tθ)^pdz )^-nm/p dθ)^-1/nm:=ℰ_p(Q,f)

where

    d_n,p(Q) := (nω_n)^1/p(nm( ))^1/nm,

and

    a_n,p=n^1/p(n-p/p-1)^p-1/p[ω_n/Γ(n)Γ(n/p)Γ(n+1-n/p)]^1/n,    a_n,1=lim_p→ 1^+a_n,p,

 
 We remark that the constant d_n,p(Q) is so that the following comparison is valid.



	Let p≥ 1,m∈, and f∈ W^1,p(). Then, for every convex body Q in ℝ^m containing the origin, one has
	
    ℰ_p(Q,f) ≤∇ f_L^p(),

 with equality when f is radially symmetric. For p=1, the inequality can be extended to BV(^n). In this case, ∇ f_L^1() should be understood as |Df|(), the total variation of f.


An immediate consequence of Theorem <ref> is that (<ref>) is stronger than the classical L^p Sobolev inequality (<ref>).  Here, note that the “multiplicity” m is hidden implicitly in the dimension of Q for the notation ℰ_p(Q,f). We remark that the energy ℰ_p(Q,f) can be viewed as a higher-order affine version of the Dirichlet energy ∇ f_L^p(), as well as the L^p affine energy ℰ_p(f). In particular, when m=1,Q=[-1/2,1/2], we have ℰ_p([-1/2,1/2], f)= ℰ_p(f); when m=1, Q=[0,1], this recovers the asymmetric L^p affine energy defined implicitly in Haberl-Schuster <cit.>. A consequence of Theorem <ref> is that 
    ℰ_p(Q,f^⋆) = ∇ f^⋆_L^p(),
 regardless of the choice of m∈ and convex body Q⊂^m containing the origin.

A natural question is: is there a Pólya-Szegö principle for the higher-order affine energy ℰ_p(Q,f)? We answer this question positively.



  Let p≥ 1,m∈, f∈ W^1,p(), and a convex body Q in ^m containing the origin. Then,
    
    ℰ_p(Q,f)≥ℰ_p(Q,f^⋆).

    When p>1 and f satisfies the minor regularity assumption 
    
    ({x:|∇ f^⋆(x)|=0}∩{x: 0<f^⋆(x)<f_L^∞()})=0,

    there is equality if and only if the level sets of f are dilations of an ellipsoid in  with respect to its center.
  
  
  Note that Theorem <ref> is valid for all p≥ 1 whereas (<ref>) is only true for p∈ [1,n).

We remark here that the regularity assumption (<ref>) for the equality condition cannot be removed due to a classical counterexample constructed by Brothers and Ziemer <cit.>. In the m=1 case, Theorem <ref> is due to Lutwak, Yang, and Zhang <cit.>, Cianchi, Lutwak, Yang, and Zhang <cit.>, Haberl, Schuster, and Xiao <cit.>, and Nguyen <cit.>. The extension to the space of BV() in this case is due to Wang <cit.>. Our methods of proof, therefore, are inevitably influenced by them.

Affine Pólya-Szegö principles can be used to “upgrade” classical (non-affine) isoperimetric inequalities. Indeed, Theorem <ref>, <ref>, in combination with  the classical L^p Sobolev inequality (<ref>), imply the mth-order affine Sobolev inequality (<ref>) established in <cit.>. In fact,

    ℰ_p(Q,f)≥ℰ_p(Q,f^⋆)= ∇ f^⋆_L^p()≥ a_n,pf^⋆_L^np/n-p() = a_n,pf_L^np/n-p().


The same philosophy will be exploited in Section <ref> to demonstrate a variety of affine versions of classical (non-affine) Sobolev-type inequalities.

Finally, in Section <ref>, we consider an associated Poincaré-type inequality for ℰ_p(Q,f) when Q is origin-symmetric.
 Let Ω⊂ be a bounded domain containing the origin in its interior, m∈, Q∈[m] be origin-symmetric, and 1 ≤ p < ∞. Then, there exists a constant C:=C(n,m,p,Q, Ω) >0 such that, for any function f ∈ W^1,p_0(Ω) it holds that 

    ℰ_p(Q,f) ≥ C f_L^p().
  
 



§ PRELIMINARIES



We first recall some rudimentary facts about convex bodies; here and throughout this section, the book <cit.> by Schneider is the standard reference. We denote by [d]⊃[d] ⊃[d], respectively, the collection of convex bodies in ^d, those that additionally contain the origin, and those that contain the origin in their interiors. We use 𝕊^d-1 for the centered unit sphere in ^d.

For each K∈[d], its polar body K^∘∈[d] is given by 

    K^∘ = ⋂_x∈ K{y∈^n: y^tx≤ 1}.

The support function h_K:𝕊^d-1→ℝ of a compact, convex set K⊂^d is given by

    h_K(u)=sup_y∈ Ky^tu,      for every  u ∈𝕊^d-1.

Note that the support function can be extended to ^d by making it 1-homogeneous. The Minkowski functional of K∈[d] is defined to be 

    y_K=inf{r>0:y∈ rK}.

Note that ·_K is the possibly asymmetric norm whose unit ball is K. The Minkowski functional, support function, and polar body are related via the following equation:

    h_K(u) = u_K^∘.

By polar coordinates, the volume of K∈[d] can be expressed in the following way through its Minkowski functional:

    [d](K)=1/d∫_θ_K^-ddθ.



The surface area measure, σ_K, for a K ∈[d] is the Borel measure on 𝕊^d given by

    σ_K(D)=ℋ ^d-1(n^-1_K(D)),

for every Borel subset D of , where n_K^-1(D) consists of all boundary points of K with outer unit normals in D. 

 Suppose the boundary ∂ K of a compact set K⊂^d has enough regularity, e.g. K is convex or C^1 smooth. The mixed volume of K with a compact, convex set L is given by

    V_1(K,L)=1/d∫_∂ Kh_L(n_K(y))dℋ^d-1(y).

Minkowski's first inequality (see, for example, <cit.>) states:

    V_1(K,L)^d≥(K)^d-1(L),

with equality if and only if K is homothetic to L.

When K∈[d], Lutwak <cit.> introduced the L^p surface area measure, denoted by σ_K,p, of K,

    dσ_K,p(u)=h_K(u)^1-pdσ_K(u).

The L^p surface area measure is arguably one of the cornerstones of the L^p Brunn-Minkowski theory that occupies much of the research in the theory of convex bodies in the last few decades.

We now collect some notations and basic facts about a measurable function f ^d →. Throughout the rest of the section, we fix p≥ 1. 

A function f is said to be non-constant if, for every α∈, ({v ∈ f(x) ≠α}) >0. A function f ^d → is said to have a weak derivative if ∇ f exists in the weak sense; that is, there exists a measurable vector map ∇ f:→ such that

    ∫_^d f(v)  divψ(v) dv = - ∫_^d (∇ f(v))^t ψ(v) dv

for every compactly supported, smooth vector field ψ:^d →^d (see <cit.>). We say a function is an L^p Sobolev function, and belongs to the L^p Sobolev space W^1,p(^d), if f,∇ f ∈ L^p(^d).  The space of functions of bounded variation, denoted by BV(^d), is very similar, except in the right-hand side of (<ref>), the vector map ∇ f is replaced with a more generic map σ_f:^d↦^d, and integration with respect to the Lebesgue measure is replaced by integration with respect to |Df|, the total variation measure of f. The two spaces are related: if f is in BV(^d) and f has a weak derivative, then, d|Df|(v)=|∇ f(v)| dv and σ_f(v)=∇ f(v)/|∇ f(v)| for |Df| almost all v∈^d.

The space of compactly supported, infinitely-times differentiable functions on ^d will be denoted by C_0^∞(^d).

Federer's coarea formula (see e.g. <cit.>) states: if f:^d→ is Lipschitz and g:^d→ [0,∞) is measurable, then, for any Borel set A⊆, one has

    ∫_f^-1(A) ∩{|∇ f|>0} g(x) d x=∫_A ∫_f^-1(t)g(y)/|∇ f(y)| d ℋ^d-1(y) d t .

Here, ℋ^d-1 denotes the (d-1)-dimensional Hausdorff measure. 

The distribution function of a measurable function f:^d→ is given by 

    μ_f(t)=[d]({x∈^d: |f(x)| >t}),

and is finite for t>0 if f∈ L^p(^d). Its decreasing rearrangement f^∗:[0,∞)→ [0,∞] is defined by

    f^∗(s)=sup{t>0:μ_f(t)>s}  for s≥ 0.

The spherically symmetric rearrangement f^⋆ of f is defined as

    f^⋆(x)=f^∗(ω_d|x|^d)   for  x∈^d.

Spherically symmetric rearrangement preserves the distribution function, that is

    μ_f=μ_f^⋆.

Moreover, for every continuous, increasing function Φ:[0,∞)↦[0,∞), one has

    ∫_Φ(|f(x)|)dx = ∫_Φ(f^⋆(x))dx.

Similarly, one may define the rearrangement of f with respect to any K∈[d]; see, for example, <cit.>. That is,
 
    f^K(x)=f^∗([d](K)x^d_K)   for  x∈^d.

Note that when K is the centered unit ball, it recovers f^⋆.

Let ν be a finite Borel measure on  that is not concentrated in any closed hemisphere, p≥ 1,m∈, and Q∈[m]. Define the (L^p,Q) cosine transform of ν, denoted by C_p,Q^m μ :→ (0,∞) as 

    (C_p,Q^m μ) (θ) = ∫_ h_Q(v^tθ)^pdμ(v).

Note that since ν is not concentrated in any closed hemisphere, the function C_p,Q^mν is a positive, continuous function on . Moreover, since h_Q is convex and p≥ 1, when extended 1-homogeneously to ^nm, the function C_p,Q^m μ is convex.

We shall use the following definition that slightly generalizes the (L^p,Q) projection body given in Definition <ref>.



Let ν be a finite Borel measure on  that is not concentrated in any closed hemisphere, p≥ 1,m∈, and Q∈[m]. The (L^p,Q) projection body of ν is the convex body in [nm]  whose support function is given by

    h_ν = (C_p,Q^m ν)^1/p.


We then have ν = (ν)^∘. When ν = σ_K,p, the L^p surface area measure of K∈[n] one has σ_K,p =  K. We also require the following definition.
 For p ≥ 1,  m ∈, and Q ∈[m], define the (L^p,Q)-LYZ projection body of non-constant f∈ W^1,p(), f to be the convex body in ^nm whose support function is given, for every θ∈,  by 

    h_ f(θ) = (∫_^nh_Q((∇ f(x))^tθ )^p dx)^1/p=h_Q(∇ f(·)^tθ)_L^p().


It will be shown in (<ref>) that f in fact contains the origin in its interior and consequently, its polar body f:=( f)^∘ is well-defined.

We remark here the link among f, ν, and K.  Lutwak, Yang, and Zhang <cit.> showed  that there exists a unique measure on the sphere ν_f,p that is not concentrated on any great hemisphere, so that the integration over ^n in (<ref>) can replaced with integration over  with respect to ν_f,p (∇ f(x) becomes merely u∈). Historically, Q=[-1/2,1/2], and thus one can apply the even L^p Minkowski problem by Lutwak <cit.> to the even part of ν_f,p to obtain an unique centrally symmetric convex body whose L^p surface area measure is the even part of ν_f,p. This origin-symmetric convex body associated to f via the even part of ν_f,p, called by Ludwig the LYZ body of f <cit.>, became a powerful tool in convex geometry; for example, Wang <cit.> used the LYZ body to characterize, independently of Nguyen, equality conditions for the Pólya-Szegö principle for the functional ℰ_p(f). This approach was later extended to another energy functional over the Grassmanian by Kniefacz and Schuster <cit.>.

In our situation, we would follow <cit.> and apply, for example, the non-even L^p Minkowski problem by Hug, Lutwak, Yang and Zhang <cit.> to the measure μ_f,p to obtain unique convex bodies ⟨ f ⟩_p∈[n], the so-called asymmetric LYZ bodies. But, the origin may be on the boundary of ⟨ f ⟩_p for 1<p < n. Consequently, f is not necessarily  applied to  ⟨ f ⟩_p, and, in particular, the Petty projection inequality (<ref>) does not readily apply. Such an issue is not present for p> n or p=1.

The following equation is immediate following Definition <ref> and the definition of ℰ_p(Q,f):

    ℰ_p(Q,f) = d_n,p(Q)(nm)^-1/nm( f)^-1/nm.





§ AN MTH-ORDER AFFINE PÓLYA-SZEGÖ PRINCIPLE



This section is dedicated to proving Theorems <ref> and <ref>, the mth-order affine Pólya-Szegö principle. We shall see in later sections that this is the source of many functional sharp affine isoperimetric inequalities. The approach adopted here is inspired by <cit.>.


 §.§ Proving Theorem <ref>

A bit of simple matrix manipulation is needed. For each T∈ GL(n), consider the map T∈ GL(nm) given as T(x_1, ⋯, x_m) = (Tx_1,…,  Tx_m) for any x_1,…, x_m∈^n. It is simple to see that if T∈ O(n), then T∈ O(nm). In particular, if θ∈, then Tθ∈. Write θ = (θ_1, …, θ_m) where θ_i∈ℝ^n. Then Tθ = (Tθ_1, …, Tθ_n) and consequently, for every v∈ℝ^n, we have

    v^t Tθ = (v^t Tθ_1, ⋯, v^tTθ_n)= ((T^tv)^tθ_1, …, (T^tv)^tθ_n)= (T^tv)^tθ.




	When integrating on O(n) with respect to dT, we mean integrating with respect to the Haar measure on O(n). We start by deriving a new formula for d_n,p(Q). Note that the integral
	
    ∫_O(n) h_Q((T^tv)^tθ)^pdT

	is independent of v∈. This and the definitions of dT and  imply
	
    d_n,p(Q)^nm   = (nω_n)^nm/p nm ()
                =  (nω_n)^nm/p∫_θ_^-nmdθ
       = ∫_(1/nω_n∫_ h_Q(v^tθ)^pdv )^-nm/pdθ
       =∫_(∫_O(n) h_Q((T^t e_1)^tθ)^pdT )^-nm/pdθ.

	Here e_1∈ is, say, the first vector in the canonical basis of ^n. Henceforth, we work with W^1,p(); the proof still works with minor modifications in the p=1,f∈ BV() case.
	
	Recall that for every T∈ O(n), we have T∈ O(nm). Thus, a change-of variable formula and (<ref>)  imply that for every T∈ O(n), we have, 
	
    (d_n,p(Q)^-1ℰ_p(Q,f))^-nm   =∫_h_Q(∇ f(·)^tθ)_L^p()^-nm dθ
    
            = ∫_h_Q(∇ f(·)^tTθ)_L^p()^-nm dθ   = ∫_h_Q((T^t∇ f(·))^tθ)_L^p()^-nm dθ.

	Consequently, by the fact that dT is a probability measure on O(n), the Fubini theorem and Jensen's inequality yield
 
    (d_n,p(Q)^-1ℰ_p(Q,f))^-nm
    			=    ∫_O(n)∫_h_Q((T^t∇ f(·))^tθ)_L^p()^-nm dθ dT
    
    		=   ∫_∫_O(n)h_Q((T^t∇ f(·))^tθ)_L^p()^-nm dT dθ
    ≥   ∫_(∫_O(n)h_Q((T^t∇ f(·))^tθ)_L^p()^pdT)^-nm/p dθ.
 
 Next, we insert the definition of h_Q((T^t∇ f(·))^tθ)_L^p(). We then use the homogeneity of support functions, the Fubini theorem once again, the fact that (<ref>) is independent of v∈, and finally (<ref>) to obtain from (<ref>)
	
    (d_n,p(Q)^-1ℰ_p(Q,f))^-nm
    ≥   ∫_(∫_O(n)∫_^n |∇ f(z)|^ph_Q((T^t ∇ f(z)/|∇ f(z)|)^tθ)^pdzdT)^-nm/p dθ
    
    		=   ∫_(∫_^n |∇ f(z)|^p∫_O(n)h_Q((T^t ∇ f(z)/|∇ f(z)|)^tθ)^pdTdz)^-nm/p dθ
    
    		=    ∇ f_L^p(^n)^-nm∫_(∫_O(n)h_Q((T^t e_1)^tθ)^pdT)^-nm/p dθ
    
    		=    ∇ f_L^p(^n)^-nm d_n,p(Q)^nm.
 
	This yields the desired inequality. When f is radially symmetric, for every T∈ O(n), by a change-of-variable and then the chain rule, we have,
	
    h_Q((T^t∇ f(·))^tθ)_L^p()^p    = ∫_^n h_Q((T^t∇ f(z))^tθ)^pdz=  ∫_^n h_Q((T^t∇ f(Tz))^tθ)^pdz
       = ∫_^n h_Q((∇ f∘ T(z))^tθ)^pdz= ∫_^n h_Q((∇ f(z))^tθ)^pdz 
       = h_Q((∇ f(·))^tθ)_L^p()^p.


	Thus, equality holds in (<ref>) by the equality condition of Jensen's inequality.
		




 §.§ Proving the inequality in Theorem <ref>

Throughout this section, let p∈ [1,∞) be fixed. We first prove an inequality for the polar projection bodies ν introduced in Definition <ref>.


	Let ν be a finite Borel measure on  that is not concentrated in any closed hemisphere. Then, there exists K∈[n] (dependent on ν and p) such that 
    
    1 = 1/n∫_ h_K(u)^p dν(u)

    and
    
    (K)h_K(u)^p-1dμ(u)=dσ_K(u).

 Moreover, the convex body K and the Borel measure ν satisfy, for every m∈ and Q∈[m],
	
    (ν ) (K)^-m≤( ) ()^nm/p -m.

    


	Since ν is not concentrated in any closed hemisphere, we may  construct a sequence of discrete measures ν_i such that ν_i are not concentrated in any closed hemisphere and ν_i converges to ν weakly. See, for example, <cit.>. 
	
	By the solution to the discrete L^p Minkowski problem (see, e.g.,  <cit.>), there exist polytopes P_i∈[n] such that 
    
    (P_i)h_P_i(u)^p-1dν_i(u)=dσ_P_i(u).

    Since each P_i∈[n], this means
	
    ν_i = (P_i)^-1σ_P_i, p.
 
	An application of <cit.> shows that P_i is uniformly bounded. Blaschke's selection theorem <cit.> now implies that, by passing to a subsequence, we may assume the P_i converge, in the Hausdorff metric, to some K∈[n]. 
 
 By (<ref>), we have
	
    1 = 1/n∫_ h_P_i(u)^p dσ_P_i, p(u)/(P_i) = 1/n∫_ h_P_i(u)^p dν_i(u) →1/n∫_ h_K^p(u) dν(u),

	where the convergence follows from the fact that h_P_i→ h_K uniformly (since h_P_i→ h_K uniformly and p≥ 1) and that ν_i converges to ν weakly. Taking the limit in (<ref>) yields (<ref>).
	
	Since each P_i contains the origin as an interior point, (<ref>) applies:
	
    ( P_i ) (P_i)^nm/p -m≤( ) ()^nm/p -m.

	The weak convergence of ν_i to ν yields
 
    (C_p,Q^m (ν_i))^1/p→(C_p,Q^m (ν))^1/p
 pointwise on  and, consequently, uniformly on . By definition, this means that
 
    ν_i →ν
 in the Hausdorff metric. By taking volume, we obtain (ν_i)→(ν). On the other hand, from (<ref>), we have that 
 
    ν_i = (P_i)^1/p P_i.

 Taking volume, this yields
	
    (P_i)^nm/p( P_i)→(ν).
	
 Letting i→∞ in (<ref>) completes the proof.

We remark that, if K ∈[n], then σ_K,p exists and, (<ref>) implies, 

    ν = (K)^-1σ_K,p.

Thus, ν = (K)^1/p K. Consequently, equation (<ref>) simply states the usual mth-order Petty projection inequality (<ref>) for this K. As another example, if ν=μ_f,p, the measure described above (<ref>), then the resultant K in Lemma <ref> is precisely ⟨ f ⟩_p.

The following lemma is critical in reducing Theorem <ref> from W^1,p() to C_0^∞(). We also obtain some useful facts about f from Definition <ref>.

	 Let p≥ 1,m∈, f_k, f∈ W^1,p(), and Q∈[m]. If f_k→ f in W^1,p(), then 
	
    θ_f_k= h_Q(∇ f_k(·)^tθ)_L^p()→ h_Q(∇ f(·)^tθ)_L^p()=θ_ f

	uniformly for θ∈, as k→∞, i.e. f_k→ f in the Hausdorff metric. Moreover, if f is not constantly 0 (up to a set of measure 0), then there exists c_0>0 such that for every θ∈
	
    θ_ f= h_Q(∇ f(·)^tθ)_L^p()>c_0,

and consequently f ∈[nm].


	Since Q∈[m], there exist M>0 such that Q⊂ MB_2^m. In particular, using the definition of support function, we get that for all x,y∈^m,
	
    |h_Q(x)-h_Q(y)|≤ M|x-y|.

Since  is compact, there exists λ>0 such that 
	
    |x^tθ|≤λ |x|,     for all  x∈^n  and all θ∈.

	
	Using (<ref>) and (<ref>), if p≥ 1 and f∈ W^1,p(),
	
    h_Q(∇ f_k(·)^tθ) - h_Q(∇ f(·)^tθ)_L^p()   ≤  M∇ (f_k-f)(·)^tθ) _L^p()
       ≤ Mλ∇ (f_k-f)_L^p()
       → 0,

	uniformly in θ∈.
	
	It remains to show (<ref>). Since Q∈[m], there exist linearly independent u_1, ⋯, u_m∈𝕊^m-1 and γ>0 such that γ u_i∈ Q. This implies,
	
    h_Q(x)≥γmax{(x^tu_1)_+, …, (x^tu_m)_+}.

	Note that for every θ∈, we have
	
    max{|θ u_1|, …, |θ u_m|}>0,

	since u_1, …, u_m are linearly independent and θ is not the zero map. This, when combined with the fact that continuous functions on a compact set achieve minima, implies the existence of τ>0 such that for all θ∈, 
	
    max{|θ u_1|, …, |θ u_m|}>τ.

	In particular, this means there exists u_i_* (for each θ) such that |θ u_i_*|>τ.
	
	 It is simple to see the map
	
    θ↦ h_Q(∇ f(·)^tθ)_L^p()

	is continuous on . Moreover, by (<ref>) and (<ref>),
	
    h_Q(∇ f(·)^tθ)_L^p()   ≥γmax_1≤ i≤ m(∇ f(·)^tθ u_i)_+_L^p()
        = γ |θ u_i_*|(∇ f(·)^tθ u_i_*/|θ u_i_*|)_+_L^p()
       ≥γτ(∇ f(·)^tθ u_i_*/|θ u_i_*|)_+_L^p()
       >0.
 
	Here, the last line follows from the first part of the proof in <cit.>. Consequently, the compactness of  implies the existence of c_0>0 such that (<ref>) holds.


We are now ready to apply the geometric inequality, Lemma <ref>, to prove Theorem <ref>. Let f∈ C_0^∞ (). Then, by Sard's theorem, one has for a.e. t>0 that 
    [f]_t:={|f|≥ t}
 is a bounded, closed set with C^1 boundary 
    ∂ [f]_t={|f|=t}.
 


We assume f is not constantly 0 (up to a set of measure 0), since otherwise, the desired inequality is trivial. 

We will prove the case when f∈ C_0^∞(). 


Recall that by Sard's theorem, for almost all t∈ (0, f_∞) we have ∇ f(y)≠ o for y∈∂ [f]_t. For those t, consider the finite Borel measure ν_t on  such that for all g∈ C(), we have

    ∫_ g(v)dν_t(v) = ∫_∂ [f]_t g(∇ f(y)/|∇ f (y)|) |∇ f(y)|^p-1 dℋ^n-1(y).

Note that ν_t is not concentrated in any closed hemisphere. This can be seen by taking g(v) = (v^tu)_+ for an arbitarily fixed u∈ and noting that the integral is positive, since f∈ C_0^∞(^n) and the outer unit normals of ∂ [f]_t form . In fact, on ∂[f]_t, one has that 

    n(y):=n_∂[f]_t(y)=∇ f(y)/|∇ f (y)|=σ_f(y).

By Lemma <ref>, there exists a convex body, which we denote by ⟨ f ⟩_t,p∈[n], such that

    (ν_t ) ≤( ) ()^nm/p -m(⟨ f ⟩_t,p)^m

and

    1 = 1/n∫_ h_⟨ f ⟩_t,p(v)^p dν_t.


 Observe that, from the coarea formula (<ref>), Minkowski's integral inequality, and (<ref>):
 
    d_n,p(Q)^-pℰ_p^p(Q, f) 
    
        =   (∫_( ∫_0^∞∫_∂[f]_t h_Q^p(n(y)^tθ) |∇ f(y)|^p-1dℋ^n-1(y)dt )^-nm/pdθ)^-p/nm
    
      =   (∫_( ∫_0^∞∫_ h_Q^p(v^tθ) dν_t(v)dt )^-nm/p dθ)^-p/nm
    
      =   (∫_( ∫_0^∞θ_ν_t^p dt )^-nm/p dθ)^-p/nm
    ≥   ∫_0^∞(∫_θ_ν_t^-nm dθ)^-p/nmdt
    
      =    (nm)^-p/nm∫_0^∞(ν_t)^-p/nm dt
    ≥    (nm())^-p/nmω_n^p-n/n∫_0^∞(⟨ f ⟩_t,p)^-p/ndt.


Inserting the definition of  d_n,p(Q) (see (<ref>)), we obtain

    ℰ_p(Q,f) ≥  n^1/pω_n^1/n(∫_0^∞(⟨ f ⟩_t,p)^-p/ndt)^1/p.

Suppose p=1. Then, one obtains from (<ref>) and (<ref>)

    1    = 1/n∫_ h_⟨ f ⟩_t,1 dν_t = 1/n∫_∂ [f]_t h_⟨ f ⟩_t,1(n(y)) dℋ^n-1(y)
        
       = V_1([f]_t,⟨ f ⟩_t,1) ≥([f]_t)^n-1/n(⟨ f ⟩_t,1)^1/n
       = μ_f(t)^n-1/n(⟨ f ⟩_t,1)^1/n,

where the last equality follows from (<ref>).  Similarly, if p>1, we have from Hölder's inequality and again (<ref>) and (<ref>)

    n^1/p(∫_∂ [f]_t|∇ f(y)|^-1dℋ^n-1(y))^p-1/p
       = (∫_∂ [f]_th_⟨ f ⟩_t,p(n(y))^p|∇ f(y)|^p-1dℋ^n-1(y))^1/p(∫_∂ [f]_t|∇ f(y)|^-1dℋ^n-1(y))^p-1/p
       ≥∫_∂ [f]_th_⟨ f ⟩_t,p(n(y))dℋ^n-1(y)
        ≥ n([f]_t)^n-1/n(⟨ f ⟩_t,p)^1/n
       = nμ_f(t)^n-1/n(⟨ f ⟩_t,p)^1/n.

In either case, we deduce that for p≥ 1

    (⟨ f ⟩_t,p)^-p/n≥μ_f(t)^p/n(n-1)(1/n∫_∂ [f]_t|∇ f(y)|^-1dℋ^n-1(y))^1-p.

Notice there is equality if and only if ⟨ f ⟩_t,p is homothetic to [f]_t for a.e. t>0. Inserting this into (<ref>), we obtain


    ℰ_p(Q,f) ≥  n^1/pω_n^1/n(∫_0^∞μ_f(t)^p/n(n-1)(1/n∫_∂ [f]_t|∇ f(y)|^-1dℋ^n-1(y))^1-pdt)^1/p.


Next, from another use of the coarea formula (<ref>), we have (see e.g. <cit.>)

    μ_f(t) = ([f]_t∩{∇ f =0})+∫_t^∞∫_{|f|=τ}|∇ f(y)|^-1dℋ^n-1(y)dτ.

Notice both terms on the right-hand side are nonincreasing. Therefore, we may differentiate in the variable t and throw-away the first term to deduce

    -μ^'_f(t) ≥∫_∂ [f]_t|∇ f(y)|^-1dℋ^n-1(y),   for a.e.  t>0.

One has, see e.g. <cit.>, that equality holds in (<ref>) if and only if f=f^⋆.

Inserting (<ref>) into (<ref>) yields

    ℰ_p(Q,f) ≥ nω_n^1/n(∫_0^∞μ_f(t)^p/n(n-1)/(-μ^'_f(t))^p-1dt)^1/p.

By concatenating all the equality cases mentioned above, equality in (<ref>)  holds if f is replaced by f^*. By the fact that μ_f(t) = μ_f^*(t), this completes the proof in the case of f∈ C_0^∞.

The general case then follows from approximation. This is essentially the same as <cit.>, but we recall the approximation for the convenience of the reader. Given f∈ W^1,p(), let {f_k} be a sequence of C_0^∞() functions that converge to f in W^1,p(). By the first part of the proof, for all k=1,2,…,

    ℰ_p(Q,f_k^*) ≤ℰ_p(Q,f_k)

Lemma <ref> and the definition of ℰ_p(Q, ·) show that 

    lim_k→∞ℰ_p(Q,f_k)=ℰ_p(Q,f).


On the other hand, since rearrangements are contractive in L^p() (see <cit.>), we know f_k^*→ f^* in L^p() and consequently f_k^⋆→ f^⋆ weakly in W^1,p(). This, when combined with Theorem <ref> and the fact that the L^p gradient norm is lower-semicontinuous with respect to weak convergence in W^1,p(), shows

    ℰ_p(Q,f)=lim_k→∞ℰ_p(Q,f_k)   ≥lim inf_k→∞ℰ_p(Q,f_k^*)= lim inf_k→∞∇ f_k^*_L^p()
       ≥∇ f^*_L^p()=ℰ_p(Q,f^*).


We save the equality conditions for Section <ref> below.





Following <cit.>, we call the sets the ⟨ f ⟩_t,p the asymmetric convexification of [f]_t.  We isolate from (<ref>) the following inequality:


    ℰ_p^p(Q, f)     > nω_n( )^p/nm∫_0^∞(ν_t)^-p/nm dt
       ≥ nω_n^p/n∫_0^∞(⟨ f ⟩_t,p)^-p/ndt.

This may be viewed as the higher-order analog of the convex Lorentz-Sobolev inequality found in <cit.>.




 §.§ The equality conditions in Theorem <ref>


This section is dedicated to proving the following statement:
     Let p> 1, f∈ W^1,p(), and Q∈[m]. Suppose f satisfies the minor regularity condition (<ref>). Then, ℰ_p(Q,f) = ℰ_p(Q,f^*) if and only if f(x)=f^E(x+x_0) for some x_0∈ and origin-symmetric ellipsoid E∈[n].

Our approach is heavily inspired by <cit.>. Our first step is to introduce some notation.

  
  Let p ≥ 1, m ∈, and fix some Q ∈[m]. Given a compact set L⊂^nm with positive volume, we define the (L^p, Q)-centroid body of L, L, to be the convex body in  with the support function 
  
    h_ L(v)^p= 1/(L)∫_L h_Q(v^tx)^p dx.


This of course extends on the classical centroid body operators, which we will discuss in more detail further below. We now derive a different formula for ℰ_p(Q,f).



Let p≥ 1,m∈, f∈ W^1,p() not identically zero, and Q∈[m]. Then,

    ℰ_p(Q,f) = d_n,p(Q) (( f)(nm+p) ∫_^nh_ f(∇ f(x))^p dx)^-1/nm.



    By definition, we have from Fubini's theorem and polar coordinates
    
    ∫_^nh_ f(∇ f(x))^p dx 
    
            =   1/( f)∫_ f∫_^n h_Q((∇ f(y)^tx)^p dy dx
            
    
            =   1/( f)∫_ fx^p_ f dx
            
    
            =   1/( f)∫_θ^p_ f∫_0^θ^-1_ fr^nm-1+pdrdθ
    
            =   1/( f)(nm+p)∫_θ^-nm_ f dθ
    
            =   1/( f)(nm+p)∫_(∫_^nh_Q((∇ f(x))^tθ )^p dx)^-nm/pdθ
    
            =   ℰ_p(Q,f)^-nmd_n,p(Q)^nm/( f)(nm+p).



Our next step is to replace ( f) with ( f). We will need the following from <cit.>. We suppress the definition of star body or compact domain; simply note that this applies to elements of [nm].

     Fix n,m ∈. 
Let L⊂^nm be a compact set with positive volume, Q ∈[m], and p ≥ 1. Then
    
    ( L)/(L)^1/m≥()/()^1/m.

If L is a star body or a compact set with piecewise smooth boundary, then there is equality if and only if L =  E up to a set of zero volume for some origin-symmetric ellipsoid E ∈[n].

This of course extends the known m=1 cases: the classical Q=[-1/2,1/2] results from Busemann and Petty <cit.> when p=1 and from Lutwak, Yang and Zhang <cit.> when p>1. By setting Q = [-α_1, α_2], α_1,α_2>0 we obtain the asymmetric L^p case by Haberl and Schuster <cit.>. 

Setting L= f in (<ref>), one obtains

    ( f)^1/m≤()^1/m/()( f),

with equality if and only if f= E for some origin-symmetric ellipsoid E∈[n]. To work with these equality conditions we use <cit.>: ∘ is bijective on the class of centered ellipsoids, in the sense that

    E = ω_n^1/p(E)^-1/p(m/ω_n(nm+p))^1/p E.

Combining this with Lemma <ref>, we obtain the following.



Let p≥ 1,m∈, f∈ W^1,p() not identically zero, and Q∈[m]. Then,
    
    ℰ_p(Q,f) ≥(ω_n/( f))^1/n(∫_^nh_ f(∇ f(x))^p dx)^1/p.

    with equality if and only if f = E for some origin-symmetric ellipsoid E∈[n].


    Observe that, by using, Lemma <ref>, (<ref>), (<ref>), and (<ref>),
    
    ℰ_p(Q,f) = ℰ_p(Q,f)^nm+p/pℰ_p(Q,f)^-nm/p
        = ℰ_p(Q,f)^nm+p/p d_n,p(Q)^-nm/p(( f)(nm+p) ∫_^nh_ f(∇ f(x))^p dx)^1/p
       =d_n,p(Q)(nm)^-nm+p/nmp(nm+p)^1/p( f)^-1/nm(∫_^nh_ f(∇ f(x))^p dx)^1/p
       ≥(ω_nnm+p/m)^1/p(()/( f))^1/n(∫_^nh_ f(∇ f(x))^p dx)^1/p.

    Using (<ref>) completes the proof.


We have the following two facts. The first can be seen as a Pólya-Szegö principle for convex symmetrization.


    Fix p>1 and origin-symmetric K∈[n]. For any f∈ W^1,p(), one has
    
    ∫_h_K(∇ f(x))^p dx ≥∫_h_K(∇ f^K(x))^p dx.

    Moreover, if f is nonnegative and satisfies the minor regularity assumption of 
    ({x:|∇ f^K(x)|=0}∩{x: 0<f^K(x)<f_L^∞(^n)})=0,

then equality holds in (<ref>) if and only if f(x)=f^K(x+x_0) for some x_0∈.



    Let p > 1, m∈, f∈ W^1,p() not identically zero, and Q∈[m]. Fix any K∈[n] such that (K)=ω_n. Then, one has
    
    ℰ_p(Q,f^⋆) = (∫_^nh_K(∇^K f(x))^p dx)^1/p.



    We use from Theorem <ref> that ℰ_p(Q,f^⋆)=∇ f^*_L^p()=ℰ_p([-1/2,1/2],f^⋆), and then the claim follows from <cit.>.

The equality characterization for Theorem <ref> now follows: we set 
    L=(ω_n/( f))^1/n f.
 Then, (L)=ω_n and from Proposition <ref> we have ℰ_p(Q,f) ≥(∫_^nh_L(∇ f(x))^p dx)^1/p,
    with equality only when L is an origin-symmetric ellipsoid E. Finally, when L=E, use Lemma <ref> and Proposition <ref> to finish the equality characterizations.



§ APPLICATIONS OF THE AFFINE PÓLYA-SZEGÖ PRINCIPLE



The central theme of this section is to demonstrate a zoo of affine “upgrades” of well-known functional isoperimetric inequalities using the affine Pólya-Szegö principle (Theorem <ref>). The key ingredient is that in the case of radially symmetric functions, the affine energy ℰ_p(Q, f) reduces to its Euclidean counterpart ∇ f_L^p(). This philosophy was demonstrated in (<ref>) when it was shown that Theorem <ref>, Theorem <ref>, and the classical L^p Sobolev inequality (<ref>), imply the mth-order L^p affine Sobolev inequality (<ref>).

The same philosophy applies to a wide range of well-known functional inequalities. We shall skip the proofs of most of the theorems to-be-presented, as they are more or less just along the lines of (<ref>). Occasionally, a proof will be given if additional details are needed. We remark that these affine inequalities will imply their classical (nonaffine) counterparts, thanks to Theorem <ref>.

Note that the L^p Sobolev inequality (<ref>) and the mth-order affine Sobolev inequality (<ref>) are only valid for p∈ [1,n). For p>n, the Morrey-Sobolev embedding theorem states that any compactly supported function in W^1,p() is essentially bounded. This can be written as a sharp inequality, see, e.g., Talenti <cit.>:
 
    f_L^∞() ≤ b_n,p(supp(f))^1/p(p/n-1)∇ f_L^p(),

for every f∈ W^1,p() with support supp(f) having finite volume. Here 

    b_n,p=n^-1/pω_n^-1/n(p-1/p-n)^p-1/p.

There is equality for functions of the form 

    f_MS(x)=a(1-|x-x_0|^p-n/p-1)_+

for some a∈ and x_0∈.

Chaining Theorems <ref>, <ref>, and (<ref>) as in (<ref>) provides the following affine version of (<ref>). 


    Fix p>n ≥ 1 and f∈ W^1,p() such that supp(f) has finite volume. Then, for every m∈ and Q∈[m]
    
    f_L^∞()≤ b_n,p(supp(f))^1/p(p/n-1)ℰ_p(Q,f).

Here, the sharp constant b_n,p is given by (<ref>). There is equality for functions of the form f_MS∘ A, where f_MS is given by (<ref>) and A∈(n).

This recovers the m=1 cases given by <cit.> when Q=[-1/2,1/2]   and <cit.> when  Q=[0,1] .

It was shown in <cit.> that lim_p→∞ d_n,p(Q) exists. We shall denote the limit as d_n,∞(Q). This motivates the following definition.


    Let f∈ W^1,∞(), m∈, and let Q∈[m]. Then, the (L^∞,Q) affine Sobolev energy of f is given by 
    
    ℰ_∞(Q,f)=d_n,∞(Q)(∫_h_Q((∇ f(·))^tθ)_L^∞()^-nm dθ)^-1/nm.


Taking the limit in (<ref>) shows the following Faber-Krahn-type inequality for ℰ_∞(Q,f). This recovers the m=1, Q=[0,1] case by Haberl, Schuster and Xiao <cit.>.


    Let m∈, Q∈[m] and f∈ W^1,∞() such that supp(f) has finite volume. Then,    
    f_L^∞() ≤ω_n^-1/n(supp(f))^1/nℰ_∞(Q,f).

Equality holds if  f is of the form

    f(x)=a(1-|A(x-x_0)|)_+

for some a∈,x_0∈ and A∈(n). 


   Note that the fact that equality holds in (<ref>) for f in (<ref>) follows from a direct computation. The inequality follows from Theorem <ref> by taking a limit. Indeed, from Fatou's lemma (and the fact that d_n,p(Q) is continuous in p),
    
    f_L^∞()   ≤ω_n^-1 / n(supp(f))^1 / nlim sup _q →∞ℰ_q(Q,f) 
       ≤ω_n^-1 / n(supp(f))^1 / nd_n,∞(Q)(lim inf _q →∞∫_h_Q((∇ f(·))^tθ)_L^q()^-nm d θ)^-1/nm
       ≤ω_n^-1 / n(supp(f))^1 / nd_n,∞(Q)(∫_lim inf _q →∞h_Q((∇ f(·))^tθ)_L^q()^-nm d θ)^-1/nm
       ≤ω_n^-1 / n(supp(f))^1 / nd_n,∞(Q)(∫_h_Q((∇ f(·))^tθ)_L^∞()^-nm d θ)^-1/nm .



Let n>1. Denote

    m_n = sup_ϕ∫_0^∞ e^ϕ(t)^n/n-1-tdt,

where the supremum is taken over all non-decreasing locally absolutely continuous functions in [0,∞) such that ϕ(0)=0 and ∫_0^∞ϕ^'(t)^ndt ≤ 1. The Moser-Trudinger inequality <cit.> states that for every function f∈ W^1,n() with 0<(supp(f))<∞, we have

    1/(supp(f))∫_e^(nω_n^1/n|f(x)|/∇ f_L^n())^n/n-1dx ≤  m_n.

  The constant nω_n^1/n is  best possible in the sense that if it were to be replaced by any other larger number, the above inequality would fail for some f∈ W^1,n() with 0<(supp(f))<∞. Carleson and Chang <cit.> showed that spherically symmetric extremals do exist for (<ref>). Theorems <ref> and <ref> now imply the following affine version; when m=1, we recover the Q=[-1/2,1/2] case from <cit.> and the Q=[0,1] case from <cit.>.
 


    Let n>1, m∈, and Q∈[m]. Then, for every f∈ W^1,n() with 0<(supp(f)) < ∞, we have
    
    1/(supp(f))∫_e^(nω_n^1/n|f(x)|/ℰ_n(Q,f))^n/n-1dx ≤  m_n.

    The constant nω_n^1/n is best possible in the sense that if it were to be replaced by any other larger number, the above inequality would fail for some f∈ W^1,n() with 0<(supp(f))<∞.




Recall that the sharp logarithmic Sobolev inequality shown in <cit.> states: for f∈ W^1,2() such that f_L^2(^n)=1, one has

    ∫_|f|^2log |f|dx ≤n/2log((2/neπ)^1/2∇ f_L^2(^n)).

Ledoux <cit.> and Del Pino and Dolbeault <cit.> established the following extension of the log-Sobolev inequality: for 1≤ p <n and f∈ W^1,p()  such that f_L^p()=1, one has

    ∫_|f|^plog |f| dx ≤n/plog(c_n,p∇ f_L^p()),

where 
    c_n,p =(p/n)^1 / p(p-1/e)^1-1 / p(Γ(1+n/2)/π^n / 2Γ(1+n(p-1)/p))^1 / n,    c_n,1=lim_p→ 1^+ c_n,p.
 As for equality conditions, when p=1, Beckner <cit.> showed that one must look beyond W^1,1() to BV(), where there is equality only for dilates of characteristic functions of centered Euclidean balls. Carlen <cit.>, for p=2, and Del Pino and Dolbeault <cit.> for 1<p<n, showed there is equality if and only if there exists a>0 and x∈ such that

    f_LS(x)=π^n / 2Γ(1+n/2)/a^n(p-1) / pΓ(1+n(p-1)/p)exp(-1/a|x-x_0|^p /(p-1)).


The following corollary includes, as special case <cit.> (m=1, Q=[0,1]). It follows from Theorems <ref>, <ref>, and (<ref>).


    Let 1≤ p <n and f∈ W^1,p(). Then, for every m∈ and Q∈[m], one has
    
    ∫_|f|^plog |f| dx ≤n/plog(c_n,pℰ_p(Q,f)).

    Here, the sharp constant c_n,p is given by (<ref>). If p=1, there is equality for dilates of characteristic functions of centered ellipsoids. On the other-hand, If 1<p<n, then there is equality for functions of the form f_LS∘ A, were f_LS is of the form (<ref>) and A∈(n).


Nash's inequality, shown in its optimal form by Carlen and Loss <cit.>, states: for f∈ L^1()∩ W^1,2(), one has

    f_L^2()(f_L^2()/f_L^1())^2/n≤β_n∇ f_L^2(),

where

    β_n^2=2(1+n/2)^1+n / 2/n λ_n ω_n^2 / n,

and λ_n is the first nonzero Neumann eigenvalue of the Laplacian -Δ on radial functions on B_2^n. In fact, let u be the associated eigenfunction. Then, there is equality in (<ref>) if and only if f is a normalized and scaled version of 

    f_N(x)=
            u(|x-x_0|) - u(1),    if |x| ≤ 1,
            
    
            0,    if |x| ≥ 1,

for some x_0∈. 

Applying Theorems <ref>, <ref>, and (<ref>), we immediately obtain the following theorem, which extends the case m=1,Q=[0,1] by Haberl, Schuster and Xiao <cit.>.

    If f∈ L^1()∩ W^1,2(), then, for every m∈ and Q∈[m], one has
    
    f_L^2()(f_L^2()/f_L^1())^2/n≤β_nℰ_p(Q,f),

    where β_n is given by (<ref>). 
    There is equality for functions of the form f_N∘ A, where f_N is of the form (<ref>) and A∈(n).


We recall that the Gagliardo-Nirenberg inequalities are precisely

    f_L^r()α_n,p(r,q)≤∇ f_L^p()^θf_L^q()^1-θ

where 1<p<n, q<r≤np/n-p, θ∈ (0,1) is so that the inequality is scale-invariant. Note that inequality (<ref>) can be deduced from the L^p Sobolev inequality (<ref>), but without the optimal constant α_n,p(r,q); this is still open in general. Clearly, (<ref>) and Nash's inequality (<ref>) are special cases. In fact, the logarithmic Sobolev inequality (<ref>) is also a limiting case. 

Del Pino and Dolbeault <cit.> at the turn of the century made a breakthrough in establishing sharp constants for a range of parameters in (<ref>). This was followed by <cit.>, where mass transport was used to give an elegant proof of the same results, among other things. Suppose again that 1<p<n and now pick q∈ (p,pn/n-p-p/n-p]. Then, set

    r=p(q-1)/p-1  and  θ=n(q-p)/(q-1)(np-(n-p)q).

Let W_0^1,p,q() denote the completion of the space of smooth compactly supported functions with respect to the norm given by

    f_p,q=∇ f_L^p() + f_L^q().

Then, it was shown <cit.> that (<ref>) holds for every f∈ W_0^1,p,q() with constant

    α_n,p(r,q)=(p q/δ)^1/r((p √(π)/q-p)(n(q-p)/p q)^1/p(Γ(δ(p-1)/p(q-p)) Γ(1+n(p-1)/p)/Γ(q(p-1)/q-p) Γ(1+n/2))^1/n)^θ,

where δ=np-q(n-p). Among the class W_0^1,p,q(), equality holds if and only if f is of the form

    f_GN(x)=a(1+b|x-x_0|^p/p-1)^-p-1/q-p

for some a∈, b>0 and x_0∈. When restricted to the class W_0^1,p,q() (with the aforementioned choices of q and r), (<ref>) interpolates between the sharp L^p Sobolev inequality (<ref>) and the logarithmic Sobolev inequality (<ref>). Indeed, the former is when q=p(n-1)/(n-p) (which yields t=1 and α_n,p(r,q)=a_n,p from (<ref>)), and the latter follows by sending q→ p from above.

Applying, like before, Theorems <ref>, <ref>, and  (<ref>), we immediately obtain the following, which extends on the case m=1, Q=[0,1] by Haberl, Schuster and Xiao <cit.>. This formally interpolates between (<ref>) and (<ref>).

    Let 1<p<n,p<q≤p(n-1)/(n-p), m∈ and let r and θ be given by (<ref>). If f∈ W_0^1,p,q()∩ W^1,p(), then, for every Q∈[m], one has
    
    f_L^r()α_n,p(r,q)≤ℰ_p(Q,f)^θf_L^q()^1-θ.

There is equality for functions of the form f_GN∘ A, there A∈(n) and f_GN is of the form (<ref>).




§ THE MTH-ORDER POINCARÉ INEQUALITY



The methods used to establish the results in this section are inspired by those in <cit.>. For Ω⊂, define the spaces L^p(Ω), W^1,p(Ω), and W_0^1,p,q(Ω) analogously to the spaces L^p(), W^1,p() and W_0^1,p,q(), respectively, but with integration over Ω.  Set W_0^1,p(Ω)=W_0^1,p,p(Ω).

The main result of this section is the next inequality.  Note that the constant c_0 might not be best possible.

 Let Ω⊂ be a bounded domain containing the origin in its interior, m∈, Q∈[m] be origin-symmetric, and p ≥ 1. Then, there exists constant c_0>0 dependent on n, p, Q, and Ω, such that for any C^1 not identically zero function f compactly supported in Ω, 
    ℰ_p(Q,f) ≥ c_0 f_L^p()^nm-1/nm∇ f_L^p()^1/nm.



For simplicity, since the constant c_0 in the desired inequality is not best, we will not try to keep track of its precise value, but rather use c_0 to denote a “constant” that depends on n, p, Q, Ω which might change from line to line.

Recall the body f from (<ref>). Since Q is symmetric, f will be too. Arguing as in the proof of Lemma <ref> to obtain equations (<ref>) and (<ref>), for each θ∈, we can find a  u_i_*∈, with θ u_i_*≠ 0 and c_0 >0 such that 

    h_ f(θ) ≥ c_0 ∇ f(·)^tθ u_i_*/|θ u_i_*|_L^p().

Applying the sharp one-dimensional Poincaré inequality given in <cit.> to the above inequality, we may write 

    h_ f(θ)    ≥ c_0 (∫_| ∇ f(z)^tθ u_i_*/|θ u_i_*||^p dz)^1/p
       = c_0 (∫_( θ u_i_*/|θ u_i_*|)^⊥∫_-∞^∞|d/dt f(y+ tθ u_i_*/|θ u_i_*|)|^p dt dy)^1/p
       ≥ c_0 f_L^p() w(Ω, θ u_i_*/|θ u_i_*|)^-1
       ≥ c_0 f_L^p().

where w(Ω, ξ) is the width of Ω in the direction of ξ and the last line follows from the fact that Ω is compact. 


The inequality (<ref>) shows that 

    f⊃ c_0 f_L^p()B_2^nm.

On the other hand, observe that 

    max_θ∈𝕊^nm-1 h_ f(θ)^p    = max_θ∈𝕊^nm-1h_Q(∇ f(·)^tθ)_L^p()^p
       ≥1/nm ω_nm∫_𝕊^nm-1∫_ h_Q(∇ f(z)^tθ)^p dz dθ
       = 1/nm ω_nm∫_ |∇ f(z)|^p ∫_𝕊^nm-1∫_O(n) h_Q((T^t ∇ f(z)/|∇ f(z)|)^tθ)^p dT dθ dz
       = 1/n^2m ω_nmω_n∫_𝕊^nm-1∫_𝕊^n-1 h_Q(u^tθ)^p du dθ∇ f_L^p()^p
       = c_0 ∇ f_L^p()^p.

(Recall here c_0 might change from line to line—it is the existence of a positive constant that we are seeking for.)

Equation (<ref>) and the fact that f is origin-symmetric imply that f contains the symmetric segment [-c_0∇ f_L^p(), c_0∇ f_L^p()]. 

Consequently we have that f contains the entire double cone 

    conv(c_0f_L^p() B_2^nm,[-c_0∇ f_L^p(), c_0∇ f_L^p()] ),

and therefore, 

    [nm]( f) ≥ c_0 f_L^p()^nm-1∇ f_L^p(),

for some c_0>0.
Finally, an application of the Blaschke-Santaló inequality (see e.g. <cit.>) yields 

    ℰ_p(Q,f)   =d_n,p(Q)(∫_( ∫_ h_Q((∇ f(z))^tθ)^pdz )^-nm/p dθ)^-1/nm
        = c_0 ( f)^-1/nm≥  c_0 [nm]( f)^1/nm
       ≥  c_0f_L^p()^nm-1/nm∇ f_L^p()^1/nm.



As an immediate corollary of Theorem <ref>, we obtain the mth-order Poincaré inequality, Theorem <ref>, by using an approximation argument and the classical L^p Poincaré inequality.


Another corollary of Theorem <ref> is the following embedding theorem.  Let Ω⊂ be a bounded domain containing the origin in its interior, m∈, Q∈[m] be origin-symmetric, and 1 ≤ p < ∞. Consider the class of functions 

    ℬ_Q,p(Ω) := {f ∈ W_0^1,p(Ω) ℰ_p(Q,f) ≤ 1}.


 For any bounded domain Ω⊂ with Lipschitz boundary containing the origin in its interior, m∈, Q∈[m] that is origin-symmetric, and 1 ≤ p < n, the set ℬ_Q,p(Ω) is compactly immersed within L^p(Ω). 


 Consider any sequence of functions f_k ∈ W^1,p_0(Ω), k =1,…, such that ℰ_p(Q,f_k) ≤ 1. If there is some sub-sequence of f_k_j that converges in the L^p-norm, then we are done. If this is not the case, then there is a positive constant c for which f_k_L^p(Ω)≥ c for all k ≥ 1. However, in this case, according to Theorem <ref>, it is necessary that the sequence of f_k is bounded in the W^1,p_0(Ω). Therefore, we may apply the Rellich-Kondarchov embedding theorem to conclude the result. 




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