An Lp-Functional Busemann–Petty Centroid Inequality

J.E. Haddad (2021) “An Lp-Functional Busemann–Petty Centroid Inequality,” International Mathematics Research Notices, Vol. 2021, No. 10, pp. 7947–7965 Advance Access Publication January 23, 2020 doi:10.1093/imrn/rnz392 An Lp-Functional Busemann–Petty Centroid Inequality Julián E. Haddad1, Carlos Hugo Jiménez2,∗and Letícia A. da Silva2 1Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, 30123-970, Belo Horizonte, Brasil and 2Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, 22451-900 Rio de Janeiro, Brasil ∗Correspondence to be sent to: e-mail: [email protected].br For a convex body K⊂Rn,letpKbe its Lp-centroid body. The Lp-Busemann–Petty centroid inequality states that vol(pK)≥vol(K), with equality if and only if Kis an ellipsoid centered at the origin. In this work, we prove inequalities for a type of functional Lr-mixed volume for 1 ≤r<nand establish, as a consequence, a functional version of the Lp-Busemann–Petty centroid inequality. 1 Introduction The study of affine isoperimetric inequalities on one side and affine Sobolev inequalities for functions on Rnon the other is connected to a great extent. The equivalence of the classical isoperimetric inequality and the classical L1-Sobolev inequality has been known for quite some time (see, e.g., [1–7]). Following this path, Zhang in [8] established the equivalence of an affine L1-Sobolev inequality with the Petty projection inequality for convex bodies. Some time after, along with Lutwak and Yang, they obtained Lp versions of this equivalence. Around the same time, these authors developed a rich theory of geometric inequalities for centroid bodies and established Lpextensions of many other fundamental notions from convex geometry, such as mixed volumes and surface area. Received August 24, 2019; Revised December 14, 2019; Accepted December 19, 2019 © The Author(s) 2020. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 7948 J. E. Haddad et al. On top of the strong connections mentioned above, other geometric inequalities of an isoperimetric flavor like the Busemann–Petty centroid inequality or Blaschke– Santaló inequality, among others, have been fundamental in the study of several inequalities of Sobolev type, like Lp-log-Sobolev, Gagliardo–Nirenberg, Sobolev trace, or weighted Sobolev inequalities (e.g., [9–14]). It is important to notice that in many of the works mentioned above, where the Busemann–Petty centroid inequality was used to recover some known results for Sobolev type inequalities, this inequality provided a more direct approach. This approach often avoided the use (in their original proofs) of other well-known tools in the area of convex geometric analysis like the solution to the Minkowski problem or the theory of mixed or dual mixed volumes. In this work, we continue with this line of research. We obtain a family of inequalities for functions on Rn, inequalities of Sobolev type, that in particular recover the Lp-Busemann–Petty centroid inequality for convex bodies in Rn. Our main inequality is presented in the form of a functional mixed volume inequality. Theorem 1.1. Let fbe a C∞function and ga continuous non-negative function, both with compact support in Rn. Then, for 1 ≤r<n,q=nr n−rand λ∈n n+p,1 ∪(1, ∞), RnRn g(y)|∇f(x),y|pdyr/p dx ≥Cn,p,λ||g|| [(n+p)(λ−1)+p]r np(λ−1) 1||g||−λr (λ−1)n λ||f||r q.(1) The sharp constant Cn,p,λis computed in Section 3. After a lengthy but straightforward computation, inequality (1) extends for fand gin the following Sobolev spaces: g∈L1 1+|x|p(Rn)=h∈L1(Rn):Rn |h(x)|(1+xp 2)dx<∞ f∈W1,q,r(Rn)=h∈Lq(Rn):∇h∈Lr(Rn), still with g≥0. These are the natural spaces to look for extremal functions of inequality (1), and equality is attained if and only if fand ghave the following forms: g(x)=aGp,λ(||Ax||2), for a.e.x f(x)=bFr(||Ax||2) for positive constants a,b,A∈GLn(R),Gp,λ:R+→Rdefined by Gp,λ(t)=⎧ ⎨ ⎩ (1+tp)1 λ−1if λ∈n n+p,1  (1−tp) 1 λ−1 +if λ>1, Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 Lp-Functional Busemann–Petty Centroid Inequality 7949 and Fr(t)=(1+tr r−1)1−r n. 2 Notions and Tools from Convex Geometry In order to show the intrinsic geometric nature of inequality (1), and in particular, its relation to the Lp-Busemann-Petty centroid inequality, let us first recall some basic definitions. A convex body is a convex set K⊂Rn, which is compact and has non-empty interior. For a convex body K, its support function hK, which uniquely characterizes it, is defined as hK(x)=max{x,y:y∈K}. If Kcontains the origin in the interior, then we also have the gauge · Kand radial rK(·) functions of Kdefined, respectively, as yK:=inf{λ>0: y∈λK},y∈Rn\{0}, rK(y):=max{λ>0: λy∈K},y∈Rn\{0}. Clearly, yK=1 rK(y). For a convex body K⊂Rnand p≥1, its Lp-moment and Lp-centroid bodies, denoted by MpKand pK, are defined by their support functions hMpK(x)p=K |x,y|pdy,andhpK(x)p=1 vol(K)cn,pK |x,y|pdy,(2) respectively, where cn,p=ωn+p ω2ωnωp−1and ωmis the m-dimensional volume of the unit ball Bof Rm.TheLp-Busemann–Petty centroid inequality states that vol(pK)≥vol(K)or vol(MpK)≥cn/p n,pvol(K) n+p p,(3) in terms of the moment body MpK. Equality holds in (3)ifandonlyifKis an 0-symmetric ellipsoid. Centroid bodies for p=1 can be found for the 1st time in a work of Blaschke [15], whereas the Busemann–Petty centroid inequality for p=1 is due to Petty [16]. The Lpversion of centroid bodies above was introduced by Lutwak and Zhang [17], while (3) was obtained by Lutwak et al. in [18]. For the history of the Busemann–Petty centroid inequality and a comprehensive introduction on centroid and moment bodies, we refer to Chapter 10 in [19]. Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 7950 J. E. Haddad et al. The theory of mixed volumes, first developed by Minkowski [20,21], is one of the pillars of the Brunn–Minkowski theory, it provides us with a unified approach to the study of several of the most important quantities from convex geometry, such as volume, mean width, surface area, among others. At the same time, it has been fundamental in many other problems ranging from characterization of special families of convex bodies to establish new isoperimetric inequalities, we refer to [19,22] for a comprehensive introduction to the theory of mixed volumes. There are several extensions of the concept of mixed volume; in this work, we will focus mainly on the dual mixed volume and the Lp extension of the mixed volume, concepts belonging to the dual and Lp-Brunn–Minkowski theory, respectively. Regarding the latter, we have the following Lpextension of mixed volume; for some background on this, we refer to [23]andto[24] and the references therein. For r≥1, the Lr-mixed volume Vr(K,L)of convex bodies Kand Lis defined by Vr(K,L)=r nlim ε→0 vol(K+rε·rL)−vol(K) ε, where K+rε·rLis the convex body defined by hK+rε·rL(x)r=hK(x)r+εhL(x)r,∀x∈Rn. One of the main aspects of mixed volumes is that they admit an integral representation. As in the classical case for the Lrversion, it is known (see [23]) that there exists a unique finite positive Borel measure Sr(K,.)on Sn−1such that Vr(K,L)=1 nSn−1 hL(u)rdSr(K,u),(4) for each convex body L. If 1 ≤r<∞and Kand Lare convex bodies in Rncontaining the origin as interior point, then we can find also in [23]that Vr(K,L)≥vol(K)n−r nvol(L)r n,(5) with equality if and only if Kand Lare dilates of each other for r>1 and if and only if Kand Lare homothetic if r=1. Combining inequalities (5)and(3), we obtain Vr(L,MpK)≥cr/p n,pvol(L)n−r nvol(K) (n+p)r np .(6) Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 Lp-Functional Busemann–Petty Centroid Inequality 7951 Taking L=MpKin (6), we recover (3); hence, (6) is an equivalent formulation for the Lp- Busemann–Petty centroid inequality. This and similar geometric inequalities for mixed volumes involving centroid and projection bodies were already considered in [25]. The main result, Theorem 1.1, is a functional version of inequality (6), replacing the sets Land Kby functions fand g. In order to establish a functional version of (6) and considering the integral representation of the geometric Lr-mixed volume (4), let us recall the following result obtained by Lutwak et al., where they introduced the concept of surface area measure of a Sobolev function. The Lr-surface area measure of a function f:Rn→Rwith Lr-weak derivatives is given by Lemma 2.1 (Lemma 4.1 of [24]). Given 1 ≤r<∞and a function f:Rn→Rwith Lr-weak derivatives, there exists a unique finite Borel measure Sr(f,.)on Sn−1such that Rn φ(−∇f(x))rdx=Sn−1 φ(u)rdSr(f,u),(7) for every non-negative continuous function φ:Rn→Rhomogeneous of degree 1. If fis not equal to a constant function almost everywhere, then the support of Sr(f,.)cannot be contained in any n−1 dimensional linear subspace. Conversely, for a convex body L, the function fL(x)=F(xL)satisfies Sr(f,.)= Sr(L,.)if Fis a smooth decreasing compactly supported function F:R+→R+satisfying ∞ 0 tn−1|F(t)|rdt=1 (see [24]). By the Sobolev inequality, we have Rn fL(x)nr n−rdx≤c nr n−r s(nωn)n n−rvol(L) ωn , where csis the sharp constant in the Sobolev inequality on Rn, and there is equality when F(t)=aFr(t)with a,b>0, where Fr(t)=(1+tr r−1)1−r n. The function F(||x||2)is an extremal function of the euclidean Lr-Sobolev inequality on Rn. Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 7952 J. E. Haddad et al. In view of identity (7), for any fand Lsuch that Sr(f,.)=Sr(L,.),wehave Vr(L,K)=1 nRn hK(−∇f(x))rdx. This motivates the following definition: Definition 2.2. Given 1 ≤r<∞and a function f:Rn→Rwith Lr-weak derivatives, we define Vr(f,K)=1 nRn hK(−∇f(x))rdx. The Lp-Sobolev inequality for general norms was proved in [26]and[27] and can be stated as a mixed volume inequality for functions as follows: Theorem 2.3. If fis a function with Lr-weak derivatives and compact support in Rn and Kis an origin-symmetric convex body, then for 1 <r<nand q=nr n−r Vr(f,K)≥cr 1fr qvol(K)r n,(8) where c1is the optimal constant and equality holds in (8)ifandonlyiff(x)=aFr(bxK) for some a,b>0. Taking f(x)=Fr(xL), we recover inequality (5). Theorem 2.3 was originally proved using an innovative approach based on optimal transportation of mass in [26]andin[27] using convex symmetrization. In Section 4, we give an alternative, simpler, and elementary proof of this inequality using the tools developed in [28]. Some of the tools we are using here, specially those contained in [24], have been used in the study of Sobolev type inequalities. Their approach is often based on a functional extension of the so-called LYZ body and other known geometric inequalities for projection and polar projection bodies (see [19, Subsection 10.15] and references therein for more on this). Let us go back to the definition of the moment body (2). It has been noticed that hMpKis a convex function regardless of the set K(see e.g., [29, Chapter 5]). The following definition has already appeared (in a slightly different way) in [30] and an asymmetric version of it in [31]. In both cases, it was used in the context of valuations to study the moment and centroid operators following the spirit of [32]. Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 Lp-Functional Busemann–Petty Centroid Inequality 7953 Definition 2.4. If gis a non-negative measurable function with compact support on Rn, then we define the convex body Mpgby hMpg(ξ)p=Rn g(x)|x,ξ|pdx. The left-hand side of (1) has the following geometric meaning: Vr(f,Mpg)=1 nRnRn g(y)|∇f(x),y|pdyr/p dx. If Kis a convex body and g(x)=G(xK)for any non-negative continuous function G:R+→Rwith compact support, it is not hard to verify using polar coordinates that Mpg=(n+p)∞ 0 tn+p−1G(t)dt1/p MpK. Our main result (Theorem 1.1) is a consequence of Theorem 2.3 and Theorem 2.5 below: Theorem 2.5. If gis a non-negative measurable function with compact support in Rn, then, for each λ∈n n+p,1 ∪(1, ∞),wehavethat vol(Mpg)p n≥cn,pan,p,λ||g|| (n+p)(λ−1)+p (λ−1)n 1||g||−λp (λ−1)n λ,(9) where an,p,λis given by the Lemma 3.4. Let Gp,λ:R+→Rbe defined by Gp,λ(t)=⎧ ⎨ ⎩ (1+tp)1 λ−1if λ<1 (1−tp) 1 λ−1 +if λ>1, then taking g(x)=Gp,λ(xK)in (9), we recover (3). Equality holds in (9)ifandonlyifg(x)=aGp,λ(|Ax|2)for any a>0andA∈ GLn(Rn). Even though Theorem 2.5 contains the geometric core of the main Theorem 1.1, the term vol(Mpg)cannot be expressed in terms of gin an elementary way, whereas this is possible for Vr(f,Mpg). This is the reason why we need to combine it with Theorem 2.3 to obtain a purely functional inequality. Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 7954 J. E. Haddad et al. Let us note that Theorem 1.1 cannot be regarded as a functional mixed volume inequality in full generality since it can only be applied to a function fand the centroid/moment body of another function g. We refer the interested reader to review the works of Milman and Rotem [33,34] where they have defined a functional extension of mixed volumes and have extended some of their main properties to a functional setting. We should finally also mention other related extensions of the Busemann– Petty centroid inequality obtained by Paouris and Pivovarov in [35] where the authors obtained randomized versions of this and other important isoperimetric inequalities. The rest of the paper is organized as follows: In Section 3, we shall prove some preliminary results, including an extension of the Lp-Busemann–Petty centroid inequality, to compact domains. Then, in Section 4, we prove Theorems 2.3 and 2.5. We hope this work will shed some more light onto the deep connection between isoperimetric and functional inequalities. 3 Preliminary Results In order to prove our main result, Theorem 1.1, we consider two cases: r=1and1< r<n.Forr=1, inequality (5) holds for more general sets. As in [8], a compact domain is the closure of a bounded open set. Lemma 3.1 ([8, Lemma 3.2]). If Mis a compact domain with piecewise C1boundary and Ka convex body in Rn,then V(M,K)n≥vol(M)n−1vol(K), with equality if and only if Mand Kare homothetic. In the same spirit, the next lemma shows that the Lp-Busemann–Petty centroid inequality remains valid for a compact domain: Lemma 3.2. If Mis a compact domain in Rn,then vol(pM)≥vol(M). (10) Equality holds in (10)ifandonlyifMis an 0-symmetric ellipsoid. Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 Lp-Functional Busemann–Petty Centroid Inequality 7955 Proof. For a compact domain Mand ξ∈Sn−1, we define the set Lξ={t∈[0, ∞):tξ∈M}. Consider δ(t)=tn n,fort≥0, and the star set SM defined by its radial function ρSM(ξ) =δ−1(μ(δ(Lξ))), where μdenotes the one-dimensional Lebesgue measure. It is easy to see that vol(SM)= vol(M). Also, let s=δ(t)=tn n,thends =tn−1dt.Forx∈Rn,wehave M |x,y|pdy=Sn−1Lξ |x,tξ|ptn−1dtdξ =Sn−1Lξ |x,ξ|ptptn−1dtdξ =Sn−1δ(Lξ) |x,ξ|p(ns)p ndsdξ =np nSn−1 |x,ξ|pδ(Lξ) sp ndsdξ. On the other hand, we have SM |x,y|pdy=Sn−1ρSM (ξ) 0 |x,tξ|ptn−1dtdξ =Sn−1ρSM (ξ) 0 |x,ξ|ptptn−1dtdξ =Sn−1δ(ρSM (ξ)) 0 |x,ξ|p(ns)p ndsdξ =np nSn−1 |x,ξ|pμ(δ(Lξ)) 0 sp ndsdξ. By the Bathtub principle (see [36, Theorem 1.14, page 28]), we have δ(Lξ) sp nds≥μ(δ(Lξ)) 0 sp nds. Therefore, M |x,y|pdy≥SM |x,y|pdy. (11) Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 7962 J. E. Haddad et al. We denote by ηSt x=∇f(x) |∇f(x)|.Since hMpg(ηSt x)=Rn g(y)ηSt x,y pdy1/p =∞ 0Ng,sηSt x,y pdyds1/p , it follows that V1(f,Mpg)=1 n∞ 0Sf,t hMpg(ηSt x)dStdt =1 n∞ 0Sf,t∞ 0Ng,sηSt x,y pdyds1/p dStdt. Write hMpNg,s(ηSt x)p=Ng,sηSt x,y pdy,then V1(f,Mpg)=1 n∞ 0Sf,t∞ 0 hMpNg,s(ηSt x)pds1/p dStdt. By the co-area formula, the Minkowski integral inequality and Lemmas 3.1,3.2, 3.5,and3.4, V1(f,Mpg)≥1 n∞ 0∞ 0Sf,t hMpNg,s(ηSt x)dStp ds1 p dt ≥1 n∞ 0∞ 0Sf,t hMpNg,s(ηSt x)dStdtp ds1 p =∞ 0∞ 0 V1(Nf,t,MpNg,s)dtp ds1 p ≥∞ 0∞ 0 vol(Nf,t)n−1 nvol(MpNg,s)1 ndtp ds1 p =∞ 0∞ 0 vol(Nf,t)n−1 ndtp vol(MpNg,s)p nds1 p =∞ 0 vol(Nf,t)n−1 ndt∞ 0 vol(MpNg,s)p nds1 p ≥c 1 p n,p∞ 0 vol(Nf,t)n−1 ndt∞ 0 vol(Ng,s)n+p nds1 p ≥c1/p n,p||f|| n n−1C−(n+p)(λ−1)+p (λ−1)np n,p,λ||g|| (n+p)(λ−1)+p (λ−1)np 1||g||−λ (λ−1)n λ.  Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 Lp-Functional Busemann–Petty Centroid Inequality 7963 Remark 4.1. Let us point out that a simpler proof of Theorem 1.1 for the case r=p can be deduced from the Lp-affine Sobolev inequality [28] and the equivalence between the Lp-Busemann–Petty centroid inequality and the Lp-Petty projection inequality (see [18]). The well-known identity for sets Vp(L,pK)=ωn vol(K) ˜ V−p(K,◦ pL), where ˜ Vp(·,·)denotes the Lp-dual mixed volume and ◦ pL,theLp-polar projection body of L, can be extended to functions as Vp(f,Mpg)=˜ V−p(g,◦ pf), where we define ˜ V−p(g,L)=Rn ||x||p Lg(x)dx and h(◦ pf,ξ)p=Rn |∇f(x),ξ)|pdx. Then an application of the dual mixed volume inequality for functions ([37, Lemma 4.1]) and the Lp-Affine Sobolev inequality (which corresponds to the Lp-Petty projection inequality for functions) gives the result. Funding This work was supported by Fundação de Amparo à Pesquisa do Estado de Minas Gerais [APQ- 01542-18 to J.E.H.]; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) [PQ- 301203/2017-2 to J.E.H., 428076/2018-1 to C.H.J., PQ 305650/2016-5 to C.H.J.]; and Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro [JCNE 236508 to C.H.J. and L.A.S.]. Acknowledgments The authors acknowledge the careful reading and useful suggestions, remarks, and comments from the anonymous referees that highly improved the presentation of this work. The authors thank M. Ludwig for pointing out several references. The third author acknowledges the support of the IFMG campus Bambui while conducting this work. Downloaded from https://academic.oup.com/imrn/article/2021/10/7947/5713440 by Universidad de Sevilla user on 21 April 2023 7964 J. E. Haddad et al. References [1] Aubin, T. “Problemes isopérimétriques et espaces de Sobolev.” J. 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