# Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality

Authors: Niufa Fang; Jin Yang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes theLp mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.


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## Body

## 1. Introduction

As we have known, Minkowski addition (the vector addition of convex bodies) is the cornerstone in the classical Brunn-Minkowski theory. Combining with volume, it leads to the Brunn-Minkowski inequality that is one of the most important results in convex geometry. The first variation of volume with respect to Minkowski addition is named the first mixed volume, and its related inequality is the Minkowski inequality. For more history and developments of the Brunn-Minkowski inequality, one may refer to the excellent survey [1]. For instance, the Prékopa-Leindler inequality [2–8] is known as the functional version of the Brunn-Minkowski inequality. In recent years, finding the functional counterparts of existing geometric results, especially for log-concave functions, has been receiving intensive attentions (see, e.g., [9–34]).In 2013, Colesanti and Fragalà [35] introduced the “Minkowski addition” and “scalar multiplication,” α·f⊕β·g (where α,β>0), of log-concave functions f and g as
(1)α·f⊕β·gx=supy∈ℝnfx−yααgyββ.We remark that a functionf:ℝn⟶0,∞ is log-concave if it has the form fx=e−ux, where u:ℝn⟶ℝ∪+∞ is convex. The total mass of f is defined as
(2)Jf=∫ℝnfxdx.Similar to the case of convex bodies, Colesanti and Fragalà [35] considered the following variational
(3)δJf,g=limt→0+Jf⊕t·g−Jft,and it is called the first variation of J at f along g. The first variation,δJf,g, includes theLpmixed volume when it restrictedfandgto the subclass of log-concave functions (see [35], Proposition 3.12).Colesanti and Fragalà’s work inspired us a natural way to extend theLp geominimal surface area for convex bodies to the class of log-concave functions. For convenience, we recall the definition of Lp geominimal surface area. For a convex body K containing the origin in its interior, its Lp geominimal surface area, GpK, is defined as (the case p=1, see Petty [36], and p>1, see Lutwak [37])
(4)ωnp/nGpK=infnVpK,QVQ∘p/n:Q∈Kon,where ωn is the volume of the unit ball in n-dimensional Euclidean space ℝn, Q∘ is the polar body of Q defined by Q∘=x∈ℝn:x,y≤1,∀y∈Q, Kon denotes the class of convex bodies in ℝn that contain the origin in their interiors, and VpK,Q is the Lp mixed volume (for detailed definition, see Section 2). The fundamental inequality for Lp geominimal surface area is the following affine isoperimetric inequality (see, e.g., [37], Theorem 3.12):
(5)GpKn≤nnωnpVKn−p,with equality if and only if K is an ellipsoid.TheLp geominimal surface area, GpK, is an important notation in the Lp Brunn-Minkowski theory, which serves as a bridge connecting affine differential geometry, relative differential geometry, and Minkowski geometry. In the past three decades, the Lp geominimal surface area has developed rapidly (see [25, 38–42] for some of the pertinent results).SinceδJf,g includes the Lp mixed volume, we extend the Lp geominimal surface area to the functional version as follows.Definition 1.
Letf:ℝn⟶0,∞ be an integral log-concave function and p>0. The Lp geominimal surface area of f is defined as
(6)cnp/nGpf=infδJf,gJg∘p/n:gisalog‐concavefunction,where cn=2πn/2, and g∘x=infy∈ℝne−x,y/gy is the polar function of g.In Lemma5, we prove that the above definition includes the Lp geominimal surface area (4) when p≥1 and restricted f,g to the subclass of log-concave functions.In order to study the functional geominimal surface area, we need the integral formula ofδJ·,·. Hence, we need some notations. We write x,y for the usual inner product of x,y∈ℝn, and ∥x∥ denotes the Euclidean normal of x∈ℝn. We say that g=e−v is an admissible perturbation for f=e−u if there exists a constant c>0 such that u∗−cv∗ is convex, where u∗y=supx∈ℝnx,y−ux is the Legendre conjugate of u. Let A′ denote the set of log-concave functions given by function f such that u=−logf belongs to
(7)L′=u∈L:domu=ℝn,u∈C+2ℝn,lim∥x∥→+∞ux∥x∥=+∞.Here,domu=x∈ℝn:ux<+∞ and
(8)L=u:ℝn⟶ℝ∪+∞∣uisconvex,domu≠∅,lim∥x∥→+∞ux=+∞.Colesanti and Fragalà ([35], Theorem 4.5) provided an integral formula for the first variation δJf,g when f,g∈A′ and g is an admissible perturbation for f. For our aims, we consider the following optimization problem:
(9)infδJf,gJg∘p/n:f,g∈A′,p>0andgisanadmissibleperturbationforf.If the extremum in (9) exists, then it is denoted by cnp/nGp1f.In Section3, we prove that for p>0 and f∈A′, if Jf is finite, then there exists a unique log-concave function f¯∈A′ such that
(10)Gp1f=δJf,f¯andJf¯∘=cn.Similar to the geometric case, the unique log-concave functionf¯ is called p-Petty functions of f and denoted by Tpf.Usingp-Petty functions, we obtain the following analytic inequality with equality conditions involving Gp1f.Theorem 2.
Supposef∈A′ and p>0. If f has its barycenter at 0 (i.e., ∫ℝnxfxdx=0), then
(11)Jfp/nGp1f≤cnp/nnJf+∫ℝnflogfdx,with equality if Tpfx=fx and fx=ce−∥Ax∥2/2 for A∈SLn and c>0.


## 2. Background

### 2.1. Functional Setting

Letu:ℝn⟶ℝ∪+∞if for everyx,y∈ℝnandλ∈0,1it satisfies
(12)u1−λx+λy≤1−λux+λuy,we say u is a convex function; let
(13)domu=x∈ℝn:ux∈ℝ.By the convexity ofu, domu is a convex set. We say that u is proper if domu≠∅. The Legendre conjugate of u is the convex function defined by
(14)u∗y=supx∈ℝnx,y−ux∀y∈ℝn.Clearly,ux+u∗y≥x,y for all x,y∈ℝn; there is an equality if and only if x∈domu and y is in the subdifferential of u at x. Hence, it can be checked that
(15)u∗∇ux+ux=x,∇ux.On the class of convex functions fromℝn to ℝ∪+∞, the infimal convolution is defined by
(16)u□vx=infy∈ℝnux−y+vy∀x∈ℝn,and the right scalar multiplication by a nonnegative real number α>0,
(17)uαx=αuxα.It was proved in [21] (Proposition 2.1) that ifu,v:ℝn⟶ℝ∪+∞ are convex functions and α>0, then
(18)u□v∗=u∗+v∗,uα∗=αu∗.The following result will be used later.Theorem 3 ([43], Theorem 10.9).
LetC be a relatively open convex set, and let f1,f2,⋯ be a sequence of finite convex functions on C. Suppose that the real number f1x,f2x,⋯ is bounded for each x∈C. It is then possible to select a subsequence of f1,f2,⋯, which converges uniformly on closed bounded subsets of C to some finite convex function f.The functional Blaschke-Santaló inequality states that letf,g be nonnegative integrable functions on ℝn satisfying
(19)fxgy≤e−x,y,∀x,y∈ℝn.Iff has its barycenter at 0, which means that ∫ℝnxfxdx=0, then
(20)∫ℝnfxdx∫ℝngxdx≤cn2,with equality if and only if there exists a positive definite matrix A and C>0 such that, a.e. in ℝn,
(21)fx=Ce−Ax,x2,gy=C−1e−A−1x,x2.


### 2.2. The First Variation of the Total Mass of Log-Concave Functions

In this paper, we set(22)L=u:ℝn⟶ℝ∪+∞∣uproper,convex,lim∥x∥→+∞ux=+∞,A=f:ℝn⟶ℝ∣f=e−u,u∈L.The total mass functional off is defined as
(23)Jf=∫ℝnfxdx.The Gaussian function(24)γx=e−∥x∥22plays within class A the role of the ball in the set of convex bodies, and Jγ=2πn/2=cn. For every A∈GLn, we write
(25)γAx=e−∥Ax∥22.From the definition of polar function and Legendre conjugate of function, we note that iff∈A, then
(26)f∘=e−φ∗.The support function of log-concave functionf=e−φ is (see [44])
(27)hfx=φ∗x.This is a proper generalization, in the sense thathχK=hK.Letf=e−u,g=e−v, and let α,β>0, then
(28)α·f⊕β·g=e−uα□vβ,which in explicit form reads
(29)α·f⊕β·gx=supy∈ℝnfx−yααgyββ.The support function ofα·f⊕β·g satisfies
(30)hα·f⊕β·gx=αhfx+βhgx.In particular,(31)hα·fx=αhfx.Letf,g∈A. The first variation of J at f along g is defined as
(32)δJf,g=limt→0+Jf⊕t·g−Jft.The existence of the above limit was proved by Colesanti and Fragalà [35], and δJf,g∈−k,+∞ with k=maxinf−logg,0Jf. In particular, for every f∈A with Jf>0, then
(33)δJf,f=nJf+∫ℝnflogfdx.The functional version of Minkowski first inequality reads as follows (see, e.g., [35], Theorem 5.1): letf,g∈A and assume that Jf>0. Then,
(34)δJf,g≥JflogJgJf+n+∫ℝnflogfdx,with equality if and only if there exists x0∈ℝn such that gx=fx−x0 for ∀x∈ℝn.LetKn denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean space ℝn. We write Kon for the set of convex bodies that contain the origin in their interiors. Let VK denote the n-dimensional volume of convex body K. The volume of the standard unit ball in ℝn is denoted by ωn=πn/2/Γn/2+1. A convex body K∈Kn is uniquely determined by its support function, which is defined as hKx=maxx,y:y∈K, where ·,· denotes the usual inner product in ℝn. The polar body of K is defined by K∘=x∈ℝn:x,y≤1,∀y∈K.For realp≥1, K,L∈Kn, and real ε>0, the Minkowski-Firey Lp combination K+pε·L is a convex body whose support function is given by
(35)hK+pε·L,·p=hK,·p+εhL,·p.TheLp mixed volume VpK,L of convex bodies K and L is defined by
(36)VpK,L=pnlimε→0+VK+pε·L−VKε.The existence of this limit is showed in [45].The following result show thatδJf,g includes the Lp mixed volume for convex bodies.Proposition 4 ([35], Proposition 3.12).
Letq∈1,+∞,p=q/q−1 and K,L∈Kon. Letu=hK∘xq/q,vx=hL∘xq/q, and f=e−u,g=e−v. Then, there exists a positive constant c=cn,q such that
(37)Jf=cn,qVK,with cn,q=qn/qΓn+q/q, and
(38)pnδJf,g=cn,qVpK,L.We setA′as the subclasses ofAgiven by the functionf such that u=logf belongs to
(39)L′=u∈L:domu=ℝn,u∈C+2ℝn,lim∥x∥→+∞ux∥x∥=+∞.For log-concave functionf=e−u, the Borel measure μf on ℝn is defined by (see [35])
(40)μf=∇u#fHn.Here,Hn is the n-dimensional Hausdorff measure. We need the fact that the barycenter of μf is the origin; i.e.,
(41)∫ℝnxμfx=0.We recall that the log-concave functiong=e−v is an admissible perturbation for log-concave function f=e−u if
(42)∃a>0:u∗−av∗is convex.Colesanti and Fragalà [35] provided an integral representation of the first variation δJ·,· (see, e.g., [35], Theorem 4.5): let f=e−u and g=e−v∈A′ and assume that g is an admissible perturbation for f. Then, δJf,g is finite and is given by
(43)δJf,g=∫domuhgxdμfx.


## 2.1. Functional Setting

Letu:ℝn⟶ℝ∪+∞if for everyx,y∈ℝnandλ∈0,1it satisfies
(12)u1−λx+λy≤1−λux+λuy,we say u is a convex function; let
(13)domu=x∈ℝn:ux∈ℝ.By the convexity ofu, domu is a convex set. We say that u is proper if domu≠∅. The Legendre conjugate of u is the convex function defined by
(14)u∗y=supx∈ℝnx,y−ux∀y∈ℝn.Clearly,ux+u∗y≥x,y for all x,y∈ℝn; there is an equality if and only if x∈domu and y is in the subdifferential of u at x. Hence, it can be checked that
(15)u∗∇ux+ux=x,∇ux.On the class of convex functions fromℝn to ℝ∪+∞, the infimal convolution is defined by
(16)u□vx=infy∈ℝnux−y+vy∀x∈ℝn,and the right scalar multiplication by a nonnegative real number α>0,
(17)uαx=αuxα.It was proved in [21] (Proposition 2.1) that ifu,v:ℝn⟶ℝ∪+∞ are convex functions and α>0, then
(18)u□v∗=u∗+v∗,uα∗=αu∗.The following result will be used later.Theorem 3 ([43], Theorem 10.9).
LetC be a relatively open convex set, and let f1,f2,⋯ be a sequence of finite convex functions on C. Suppose that the real number f1x,f2x,⋯ is bounded for each x∈C. It is then possible to select a subsequence of f1,f2,⋯, which converges uniformly on closed bounded subsets of C to some finite convex function f.The functional Blaschke-Santaló inequality states that letf,g be nonnegative integrable functions on ℝn satisfying
(19)fxgy≤e−x,y,∀x,y∈ℝn.Iff has its barycenter at 0, which means that ∫ℝnxfxdx=0, then
(20)∫ℝnfxdx∫ℝngxdx≤cn2,with equality if and only if there exists a positive definite matrix A and C>0 such that, a.e. in ℝn,
(21)fx=Ce−Ax,x2,gy=C−1e−A−1x,x2.


## 2.2. The First Variation of the Total Mass of Log-Concave Functions

In this paper, we set(22)L=u:ℝn⟶ℝ∪+∞∣uproper,convex,lim∥x∥→+∞ux=+∞,A=f:ℝn⟶ℝ∣f=e−u,u∈L.The total mass functional off is defined as
(23)Jf=∫ℝnfxdx.The Gaussian function(24)γx=e−∥x∥22plays within class A the role of the ball in the set of convex bodies, and Jγ=2πn/2=cn. For every A∈GLn, we write
(25)γAx=e−∥Ax∥22.From the definition of polar function and Legendre conjugate of function, we note that iff∈A, then
(26)f∘=e−φ∗.The support function of log-concave functionf=e−φ is (see [44])
(27)hfx=φ∗x.This is a proper generalization, in the sense thathχK=hK.Letf=e−u,g=e−v, and let α,β>0, then
(28)α·f⊕β·g=e−uα□vβ,which in explicit form reads
(29)α·f⊕β·gx=supy∈ℝnfx−yααgyββ.The support function ofα·f⊕β·g satisfies
(30)hα·f⊕β·gx=αhfx+βhgx.In particular,(31)hα·fx=αhfx.Letf,g∈A. The first variation of J at f along g is defined as
(32)δJf,g=limt→0+Jf⊕t·g−Jft.The existence of the above limit was proved by Colesanti and Fragalà [35], and δJf,g∈−k,+∞ with k=maxinf−logg,0Jf. In particular, for every f∈A with Jf>0, then
(33)δJf,f=nJf+∫ℝnflogfdx.The functional version of Minkowski first inequality reads as follows (see, e.g., [35], Theorem 5.1): letf,g∈A and assume that Jf>0. Then,
(34)δJf,g≥JflogJgJf+n+∫ℝnflogfdx,with equality if and only if there exists x0∈ℝn such that gx=fx−x0 for ∀x∈ℝn.LetKn denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean space ℝn. We write Kon for the set of convex bodies that contain the origin in their interiors. Let VK denote the n-dimensional volume of convex body K. The volume of the standard unit ball in ℝn is denoted by ωn=πn/2/Γn/2+1. A convex body K∈Kn is uniquely determined by its support function, which is defined as hKx=maxx,y:y∈K, where ·,· denotes the usual inner product in ℝn. The polar body of K is defined by K∘=x∈ℝn:x,y≤1,∀y∈K.For realp≥1, K,L∈Kn, and real ε>0, the Minkowski-Firey Lp combination K+pε·L is a convex body whose support function is given by
(35)hK+pε·L,·p=hK,·p+εhL,·p.TheLp mixed volume VpK,L of convex bodies K and L is defined by
(36)VpK,L=pnlimε→0+VK+pε·L−VKε.The existence of this limit is showed in [45].The following result show thatδJf,g includes the Lp mixed volume for convex bodies.Proposition 4 ([35], Proposition 3.12).
Letq∈1,+∞,p=q/q−1 and K,L∈Kon. Letu=hK∘xq/q,vx=hL∘xq/q, and f=e−u,g=e−v. Then, there exists a positive constant c=cn,q such that
(37)Jf=cn,qVK,with cn,q=qn/qΓn+q/q, and
(38)pnδJf,g=cn,qVpK,L.We setA′as the subclasses ofAgiven by the functionf such that u=logf belongs to
(39)L′=u∈L:domu=ℝn,u∈C+2ℝn,lim∥x∥→+∞ux∥x∥=+∞.For log-concave functionf=e−u, the Borel measure μf on ℝn is defined by (see [35])
(40)μf=∇u#fHn.Here,Hn is the n-dimensional Hausdorff measure. We need the fact that the barycenter of μf is the origin; i.e.,
(41)∫ℝnxμfx=0.We recall that the log-concave functiong=e−v is an admissible perturbation for log-concave function f=e−u if
(42)∃a>0:u∗−av∗is convex.Colesanti and Fragalà [35] provided an integral representation of the first variation δJ·,· (see, e.g., [35], Theorem 4.5): let f=e−u and g=e−v∈A′ and assume that g is an admissible perturbation for f. Then, δJf,g is finite and is given by
(43)δJf,g=∫domuhgxdμfx.


## 3. FunctionalLp Geominimal Surface Areas

Analogy to convex bodies, forf∈A and p∈ℝ, we consider the following optimization problem:
(44)cnp/nGλf=infδJf,gJg∘p/n:g∈A.The following result shows that the above optimization problem includes Lutwak’sLp geominimal surface areas for convex bodies (4) when p>1 (up to a constant which is dependent onnandp). This is one of the reasons why Gpf is called the Lp geominimal surface area for log-concave function f.Lemma 5.
Letp>1, q=p/p−1, and K∈Kon. Iff=e−hK∘xq/q, then
(45)Gpf=αn,pGpK,with αn,p=1/pcn,qcn,pp/nωnp/ncn−p/n for cn,q=qn/qΓn+q/q.Proof.
LetK,L∈Kon, ux=hK∘xq/q,vx=hL∘xq/q, and f=e−u,g=e−v. It is not hard to see that
(46)v∗x=hLxpp.Then, Proposition 4 tells us that(47)Jg∘=cn,pVL∘,pnδJf,g=cn,qVpK,L,with cn,q=qn/qΓn+q/q.From the definitions ofLp geominimal surface area of convex bodies (4) and log-concave functions (44), we have
(48)cnp/nGpf=infδJf,gJg∘p/n:g∈A=infnpcn,qcn,pp/nVpK,LVL∘p/n:K,L∈Kon=1pcn,qcn,pp/nωnp/nGpK.Since we need the integral representation of the first variationδJf,g in dealing the problem (44), we focus on
(49)cnp/nGp1f=infδJf,gJg∘p/n:g∈A′andgisanadmissibleperturbationforffor f∈A′ and p∈ℝ. Trivially, Gpf≤Gp1f.We need the next lemma.Lemma 6.
Letf,g∈A′ and assume that g is an admissible perturbation for f. IfA∈SLn, then
(50)δJf∘A,g∘A=δJf,g.Proof.
Letfx=e−ux and gx=e−vx. We note that
(51)v∘A∗x=supy∈ℝnx,y−vAy=supy∈ℝnA−tx,Ay−vAy=v∗A−tx.Since∇xu∘A=At∇Axu, we have
(52)δJf∘A,g∘A=∫ℝnv∘A∗∇u∘Axf∘Axdx=∫ℝnv∘A∗At∇AxuAxfAxdx=∫ℝnv∗∇AxuAxfAxd=∫ℝnv∗∇uzfzdz=δJf,g.The following result shows that the functional geominimal surface area is affine invariant.Lemma 7.
Supposef∈A′ and p>0. IfA∈SLn, then
(53)Gp1f∘A=Gp1f.Proof.
By (51) and the definition of polar function (26), we have
(54)Jg∘A∘=Jg∘∘A−t=Jg∘,for A∈SLn. Combing with Lemma 6, we have
(55)δJf∘A,gJg∘p/n=δJf,g∘A−1Jg∘A−1∘p/n.
Therefore, we obtain(56)Gp1f∘A=Gp1f,for A∈SLn.The following lemma was proved by Cordero-Erausquin and Klartag ([46], Lemma 16).Lemma 8.
Letμ be a finite Borel measure in ℝn, and let K be the interior of convSuppμ. If x0∈K and the barycenter of μ lies at the origin, then there exists a constant Cμ,x0>0 with the following property: for any nonnegative, μ-integrable, convex function φ:ℝn⟶ℝ∪+∞,
(57)φx0≤Cμ,x0∫ℝnφdμ.The next proposition shows that the infimum in the definition of thep-geominimal surface area of log-concave function is a minimum.Proposition 9.
Letp>0 and f∈A′. IfJf is finite, then there exists a unique log-concave function f¯∈A such that
(58)Gp1f=δJf,f¯andJf¯∘=cn.Proof.
From the definition ofGp1f, there exists a sequence gi∈A′ such that Jgi∘=cn, with δJf,γ≥δJf,gi for all i, and
(59)δJf,gi⟶Gp1f.
Letgix=e−vix, then
(60)δJf,γ≥δJf,gi=∫ℝnvi∗xdμfx.First, we assume thatvi are nonnegative and vi0=0 for all i. In this case, from (14), we have
(61)vi∗x=supy∈ℝny,x−viy≥0,x−vi0=0,and
(62)vi∗0=supy∈ℝny,0−viy=−infy∈ℝnviy=0.LetKbe the interior ofconvSuppμf. By Lemma 8 and ((59)), we conclude thatvi∗are uniformly upper bound which is dependent only onf. According to Theorem 3, there exists a subsequence vij∗j=1,2,⋯ that converges pointwise in K to a convex function v∗:K⟶ℝ. We extend the definition of v∗ by setting v∗x=+∞ for x∈K¯ and for x∈∂K,
(63)v∗x=limλ→1−v∗λx.This limit always exists in0,+∞, since the function λ↦v∗λx is nondecreasing for λ∈0,1 following from the convexity of v∗ and v∗0=0. Moreover, we have that v∗λxZv∗x as λ⟶1− for any x∈K¯. Because vi∗⟶v∗ is equivalent to vi⟶v (here, v=v∗∗), hence, there exits a log-concave function f¯=e−v which satisfies the claim.In the general case, there existx0i∈ℝn and infx∈ℝnvix=di∈ℝ such that v¯ix=vix−x0i−di are nonnegative and v¯i0=0 for all i=1,2,⋯. The convexity of vi and e−vi∈A′ ensures the finiteness of di; i.e.,∣di∣<k for some k>0. Similar to the first case, we have
(64)δJf,γ≥δJf,g¯i=∫ℝnv¯i∗xdμfx,where g¯i=e−vi. Lemma 8 deduces that
(65)δJf,γ≥δJf,g¯i=∫ℝnv¯i∗xdμfx≥1Cμf,x0v¯i∗x0holds for x0∈K. Moreover,
(66)v¯i∗x=supy∈ℝny,x−v¯iy=supy∈ℝny,x−viy−x0i+di=supy∈ℝny+x0i,x−viy+di=vi∗x+x,x0i+di.Therefore,(67)Cμf,x0δJf,γ≥v¯i∗x0=vi∗x0+x0,x0i+di,i.e.,
(68)vi∗x0≤Cμf,x0δJf,γ−x0,x0i−di,for any x0∈K. Then, along the same line of the first case, we conclude that the claim of this proposition holds.The uniqueness of the minimizing function is demonstrated as follows. Supposeh1,h2∈A, such that Jh1∘=Jh2∘=cn, and
(69)δJf,h1Jh1∘p/n=infδJf,gJg∘p/n:g∈A′=δJf,h2Jh2∘p/n,i.e.,
(70)δJf,h1=δJf,h2.Leth1=e−v1 and h2=e−v2. Define h∈A′, by
(71)h=12·h1⊕12·h2=e−v11/2□v21/2.Then, from (18) and (70), we have
(72)δJf,h=∫ℝnv112□v212∗xdμfx=12∫ℝnv1∗xdμfx+12∫ℝnv2∗xdμfx=12δJf,h1+12δJf,h2=δJf,h1=δJf,h2,and by the basic inequality ab≤a+b/2 for a,b>0 and (18), we have
(73)Jh∘=∫ℝne−v11/2□v21/2∗dx=∫ℝne−1/2v1∗x+1/2v2∗xdx≤12Jh1∘+12Jh2∘,with equality if and only if h1∘=h2∘. Therefore,
(74)δJf,hJh∘p/n≤δJf,h1Jh1∘p/nis the contradiction that would arise if it was the case that h1≠h2.The unique function whose existence is guaranteed by Proposition9 will be denoted by Tpf, and will be called the p-Petty body of log-concave function f (or the λ-Petty function). The polar function of Tpf will be denoted by Tp∘f, rather than Tpf∘. For f∈A and p>0, the log-concave function Tpf is defined by
(75)Gp1f=δJf,Tpf,JTp∘f=cn.Lemma 10.
Ifp>0 and f∈A, then for A∈SLn,
(76)Tpf∘A=Tpf∘A.Proof.
From the definition ofTp and Lemma 7,
(77)δJf,Tpf=Gp1f=Gp1f∘A=δJf∘A,Tpf∘A,Lemma6 deduces
(78)δJf,Tpf=δJf∘A,Tpf∘A=δJf,Tpf∘A∘A−1.The uniqueness of Proposition9 ensures that Tpf=Tpf∘A∘A−1.By the Blaschke-Santaló inequality, we obtain the following affine isoperimetric inequality for the functional geominimal surface area.Theorem 11.
Letf∈A′ and p>0. If f has its barycenter at 0, then
(79)Jfp/nGp1f≤cnp/nnJf+∫ℝnflogfdx,with equality if Tpfx=fx and fx=ce−∥Ax∥2/2 for A∈SLn and c>0.Proof.
Takingg=f in (49), together with (33), we have,
(80)cnp/nGp1f≤δJf,fJf∘p/n=nJf+∫ℝnflogfdxJf∘p/n,i.e.,
(81)cnGp1fnn+1/Jf∫ℝnflogfdxnJfn−p1/p≤JfJf∘.
By Blaschke-Santaló inequality (20) and the above inequality, we have
(82)Gp1fnn+1/Jf∫ℝnflogfdxnJfn−p1/p≤cn.This is the desired inequality.To obtain the equality condition, first assume thatTpf=f. Formula (77) tells us that
(83)Gp1f=δJf,fandJf∘=cn.This shows that there is equality in (81). From the condition of Blaschke-Santaló inequality, we known that there exists a positive definite matrix A and c>0 such that, a.e. in ℝn,
(84)fx=ce−∥Ax∥22.Therefore, we obtain the equality condition, namely,Tpf=f and fx=ce−∥Ax∥2/2.



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*Source: 3039598-2020-07-13.xml*