Empirical forms of the Petty projection inequality

Affine isoperimetric inequalities concern functionals on classes of sets in which ellipsoids play an extremal role. Typically such inequalities involve convex bodies, taken modulo affine (or linear) transformations, and are strictly stronger than their Euclidean counterparts. The standard isoperimetric inequality can be derived from several different affine strengthenings. Such affine inequalities have come to form an integral part of convex geometry and have been extensively investigated within Brunn-Minkowski theory; see the expository survey [22] and books [9, 34] for foundational work on this subject.

A fundamental example is the Petty projection inequality. Recall that the projection body $\Pi(K)$ of a convex body $K$ in $\mathbb{R}^{n}$ is defined as follows: given a direction $\theta$ on the sphere $S^{n-1}$, the support function of $\Pi(K)$ is the volume of orthogonal projection of $K$ onto $\theta^{\perp}$ (see §2 for precise definitions). We write $\Pi^{\circ}(K)$ for the polar of the projection body. Petty’s inequality states that among all convex bodies of the same volume, ellipsoids maximize the volume of the polar projection body. Formally, it can be stated as

$$|\Pi^{\circ}(K)|\leq|\Pi^{\circ}(K^{*})|$$

(1)

where $K^{*}=rB_{2}^{n}$ is the centered Euclidean ball with radius $r$ chosen to satisfy $|K^{*}|=|rB_{2}^{n}|$. The polar projection operator $\Pi^{\circ}$ satisfies $\Pi^{\circ}(TK)=T\Pi^{\circ}(K)$ for any volume-preserving affine transformation $T$, which explains the use of ‘affine’ in this context.

Projection bodies are an important class of convex bodies in geometry and functional analysis [2, 3, 14, 35]. The volume of $\Pi^{\circ}(K)$ is related to the surface area $S(K)$ via

$$\omega_{n}^{1/n}|\Pi^{\circ}(K)|^{-\frac{1}{n}}\leq S(K),$$

where $\omega_{n}=|B_{2}^{n}|$; the latter follows directly from Cauchy’s formula and Hölder’s inequality (see [34, Remark 10.9.1]). Thus Petty’s inequality implies the classical isoperimetric inequality for convex sets. Up to normalization, the surface area $S(K)$ is one of the quermassintegrals of $K$, while the quantity $|\Pi^{\circ}(K)|^{-1/n}$ is an affine quermassintegral of $K$. Alexandrov’s inequalities state that among convex bodies of a given volume, all quermassintegrals are minimized on balls (see [34, §7.4]) . In a recent breakthrough [27], E. Milman and Yehudayoff proved that all affine quermassintegrals are minimized on ellipsoids, verifying a long-standing conjecture of Lutwak. This result establishes a family of affine inequalities that interpolate between the Petty projection inequality and the fundamental Blaschke-Santaló inequality for the volume of the polar body of $K$. The latter is equalivalent to the affine isoperimetric inequality, see e.g., [22].

Petty originally built on work of Busemann concerning the expected volume of random simplices in convex bodies, and established what is known as the Busemann-Petty centroid inequality [30]. He connected the latter to projection bodies [31] by using an inequality about mixed volumes, known as Minkowski’s first inequality ([34, §7.2]), which asserts that

$$V_{1}(K,L)\geq|K|^{n-1}|L|.$$

This idea was further developed by Lutwak and plays an important role in kindred inequalities (see [22]).

Since Petty’s seminal work in 1972, his inequality has been proven by a number of different methods, e.g., [22, 33, 26, 27]. Moreover, several generalizations of the inequality have been established. In particular, Lutwak, Yang and Zhang introduced $L_{p}$ and Orlicz versions of the projection body and proved the corresponding Petty inequalities [24, 25]. In [11, 1], a generalization to Minkowski valuations was obtained (see also [18] for a characterization of the projection body operator). Another generalization, established by Lutwak involves the notion of mixed projection bodies. Let $K_{1},\cdots,K_{n-1}$ be convex sets in $\mathbb{R}^{n}$. The support function of the mixed projection body $\Pi(K_{1},\cdots,K_{n-1})$ in a direction $\theta$ is defined as the following mixed volume:

$$h_{\Pi(K_{1},\ldots,K_{n-1})}(\theta)=nV(K_{1},\ldots,K_{n-1},[0,\theta]),$$

(2)

where $[0,\theta]$ is the line segment joining the origin and $\theta$. Lutwak established several inequalities for mixed projection bodies, one of which gives Petty’s projection inequality as a special case [19, Theorem 3.8]; namely,

$$\lvert\Pi^{\circ}(K_{1},\ldots,K_{n-1})\rvert\leq\lvert\Pi^{\circ}(K_{1}^{*},\ldots,K_{n-1}^{*})\rvert.$$

(3)

Recent active investigation around the notion of the projection body with respect to other measures and generalizations appear in [16, 15].

In this note we establish empirical versions of the Petty projection inequality and its generalizations for mixed projection bodies. The study of empirical versions of affine isoperimetric inequalities for centroid bodies and their $L_{p}$-analogues was initiated by the first two authors in [28] and further developed in [6]. A number of inequalities in Brunn-Minkowski theory have been shown to have stronger empirical forms [29], but Petty’s projection inequality has eluded our previous efforts. Inspired by recent results of E. Milman and Yehudayoff [27], and also by the approach of Campi and Gronchi in [4], our work here is intended to fill this gap.

Our main results concern randomly generated sets, obtained as linear images of a compact, convex set $C\subseteq\mathbb{R}^{m}$ under an $n\times m$ random matrix $\bm{X}$. Namely, we will consider sets of the form

$$\bm{X}C=\left\{c_{1}X_{1}+\ldots+c_{N}X_{N}:(c_{j})\in C\right\},$$

where $X_{1},\ldots,X_{m}$ are independent random vectors distributed according to densities of continuous probability distributions on $\mathbb{R}^{n}$. We will write ${\bm{X}}^{\scriptscriptstyle\#}$ for the $n\times m$ random matrix that has independent columns distributed according to $f^{*}$, the symmetric decreasing rearrangement of $f$ (see § 2.3).

More generally, it will be convenient to work with matrices $\bm{X}$ whose column vectors are grouped into blocks. Assume that $\{X_{ij}\}$ is a collection of independent random vectors such that $X_{ij}$ is distributed according to $f_{ij}$, $i=1,\ldots,n$, $j=1,\ldots,m_{i}$, where $m_{i}\geq 1$. For $\ell=1,\ldots,n$, we write $m(\ell)=m_{1}+\cdots+m_{\ell}$, and form $\bm{X}=[\bm{X}_{1}\ldots\bm{X}_{\ell}]$ with $n\times m_{i}$ blocks $\bm{X}_{i}=[X_{i1}\ldots X_{im_{i}}]$. We adopt a similar convention for ${\bm{X}}^{\scriptscriptstyle\#}$, which consists of $n\times m_{i}$ blocks $\bm{X}_{i}^{\scriptscriptstyle\#}=[X_{i1}^{*}\cdots X_{im_{i}}^{*}]$, where $X_{ij}^{*}$ are independent and distributed according to $f_{ij}^{*}$. For ease of reference, we summarize this notation in Table 1.

With this notation, our first main result concerns mixed projection bodies of random sets generated by $\bm{X}$ and $\bm{X}^{\scriptscriptstyle\#}$.

A special case of central importance concerns the classical projection body operator. Taking $\ell=1$ and writing $m=m(1)=m_{1}$ and $C=C_{1}=\ldots=C_{n}$, we have the following consequence.

Theorem 1.2 extends the Petty projection inequality (1) in various ways. Indeed, let $K$ be a convex body in $\mathbb{R}^{n}$ and let $X_{1},\ldots,X_{m}$ be independent random vectors drawn uniformly from $K$. We denote their convex hull by

$$[K]_{m}=\mathop{\rm conv}\{X_{1},\ldots,X_{m}\}.$$

In matrix notation, we have $[K]_{m}={\bm{X}}S_{m}$, where ${\bm{X}}=[X_{1}\dots X_{m}]$ and $S_{m}$ is the simplex $S_{m}:=\mathop{\rm conv}\{e_{1},\dots,e_{m}\}$. Thus if $\nu$ is Lebesgue measure, the above theorem states that

$$\mathbb{E}|\Pi^{\circ}([K]_{m})|\leq\mathbb{E}|\Pi^{\circ}([K^{*}]_{m})|.$$

(4)

Note that $\Pi^{\circ}([K^{*}]_{m})$ is not a ball and the above statement does not follow from Petty’s inequality. However, when $m\rightarrow\infty$, we get that $[K]_{m}\rightarrow K$, which implies $\Pi^{\circ}([K]_{m})\rightarrow\Pi^{\circ}(K)$, hence (1) follows from (4). The inequality $\nu\left(\Pi^{\circ}(K)\right)\leq\nu\left(\Pi^{\circ}(K^{*})\right)$ that we get from Theorem 1.2 as $m\to\infty$ can also be directly obtained by adapting the proof of Petty’s inequality in [27, Section 8.2].

More generally, let $K$ and $L$ be convex bodies in $\mathbb{R}^{n}$ and assume that the columns of ${\bm{X}}_{1}$ and ${\bm{X}}_{2}$ are distributed according to $\frac{1}{\lvert K\rvert}\mathds{1}_{K}$ and $\frac{1}{\lvert L\rvert}\mathds{1}_{L}$. For $p_{1},p_{2}\geq 1$, we define $[K]_{m_{1}}^{p_{1}}=\bm{X}_{1}B_{p_{1}}^{m_{1}}$ and $[L]_{m_{2}}^{p_{2}}=\bm{X}_{2}B_{p_{2}}^{m_{2}}$, we have

$\displaystyle\mathbb{E}\nu(\Pi^{\circ}([K]_{m_{1}}^{p_{1}}+_{p}[L]_{m_{2}}^{p_{2}}))\leq\mathbb{E}\nu(\Pi^{\circ}([K^{*}]_{m_{1}}^{p_{1}}+_{p}[L^{*}]_{m_{2}}^{p_{2}})),$

(5)

where $+_{p}$ denotes $L_{p}$-addition of sets $p\geq 1$ (see § 2); in fact, we can accommodate more general $M$-addition and Orlicz addition operations (see § 2). In a similar manner, Theorem 1.1 implies

$$\mathbb{E}\nu(\Pi^{\circ}([K]_{m_{1}}^{p_{1}},\ldots,[K]_{m_{n-1}}^{p_{n-1}}))\leq\mathbb{E}\nu(\Pi^{\circ}([K^{*}]_{m_{1}}^{p_{1}},\dots,[K^{*}]_{m_{n-1}}^{p_{n-1}}))$$

(6)

where we used the same notation as above. When $\nu$ is the Lebesgue measure, then (6) can be seen as a local version of (3) for natural families of random convex sets associated to $K_{1},\ldots,K_{n-1}$.

Further specializing to the case when $p_{1}=\ldots=p_{n-1}=\infty$, we get a corollary for the mixed projection body of centroid bodies. Recall that the centroid body $Z(L)$ of a convex body $L$ in $\mathbb{R}^{n}$ is defined by its support function via

$\displaystyle h_{Z(L)}(u)=\frac{1}{\lvert L\rvert}\int_{L}\lvert\langle x,u\rangle\rvert dx.$

The Busemann-Petty centroid inequality mentioned above is a sharp extremal inequality for the volume of $Z(K)$, which heavily influenced affine isoperimetric principles [22] and the development of $L_{p}$-Brunn–Minkowski theory [24, 25].

A stochastic notion of centroid bodies was developed in [28], which defines a random variant $Z_{m}(L)$ of $L$ as the body with support function

$\displaystyle h_{Z_{m}(L)}(u)=\frac{1}{m}\sum_{i=1}^{m}\lvert\langle X_{i},u\rangle\rvert,$

where $X_{1},\ldots,X_{m}$ are independent and identically distributed according to the normalized Lebesgue measure on $L$. Our next result concerns mixed projection bodies of independent empirical centroid bodies $Z_{m}(K_{1}),\ldots,Z_{m}(K_{n-1})$ whose support function is given by

$\displaystyle h_{\Pi(Z_{m}(K_{1}),\ldots,Z_{m}(K_{n-1}))}(y)$

$\displaystyle=nV(Z_{m}(K_{1}),\ldots,Z_{m}(K_{n-1}),[0,y]).$

We note that when $m\to\infty$,

$$V(Z_{m}(K_{1}),\ldots,Z_{m}(K_{n-1}),[0,y])\to\tilde{c}_{n}\int_{K_{1}}\cdots\int_{K_{n-1}}\lvert\mathop{\rm det}[x_{1},\ldots x_{n-1},y]\rvert dx_{1}\ldots dx_{n-1}$$

(7)

where we use [34, eq. (5.81)]. Haddad recently established a family of isoperimetric inequalities for a new class of convex bodies [12] that are defined using similar determinantal expressions as in (7) and their $L_{p}$-generalizations. Our work shows that such bodies arise naturally as limiting cases of mixed projection bodies of random sets in Theorem 1.1 when the $C_{i}$’s are chosen to be cubes.

We also present an alternate empirical version of Petty’s projection inequality. This approach is inspired by the proof of the inequality based on Busemann-Petty centroid inequality and Minkowski’s first inequality [22, 34]. We will use an empirical approximant of centroid bodies, defined as follows. For each convex body $L$ in $\mathbb{R}^{n}$, we use the notation $[L]_{m_{2}}^{\infty}=\bm{X}_{2}B_{\infty}^{m_{2}}$; this is nothing but the Minkowski sum of $m_{2}$ random segments $[-X_{2j},X_{2j}]$, where $X_{21},\ldots,X_{2m_{2}}$ are independent random vectors sampled according to $\frac{1}{\lvert L\rvert}\mathds{1}_{L}$, i.e.,

$$[L]_{m_{2}}^{\infty}=\sum_{j=1}^{m_{2}}[-X_{2j},X_{2j}].$$

Note that $\frac{1}{m}[L]_{m}^{\infty}=Z_{m}(L)$. Using this notation with $L=\Pi^{\circ}(K)$, we establish a sharp extremal inequality for the following quantity:

	$\displaystyle\mathbb{E}V_{1}([K]_{m_{1}},[\Pi^{\circ}(K)]_{m_{2}}^{\infty})$
	$\displaystyle=\frac{1}{\lvert K\rvert^{m_{1}}}\frac{1}{\lvert\Pi^{\circ}(K)\rvert^{m_{2}}}\int\limits_{K^{m_{1}}}\int\limits_{(\Pi^{\circ}(K))^{m_{2}}}V_{1}\left(\mathop{\rm conv}\{x_{11},\ldots,x_{1m_{1}}\},\sum_{j=1}^{m_{2}}[-x_{2j},x_{2j}]\right)dx_{11}\ldots dx_{1m_{1}}dx_{21}\ldots dx_{2m_{2}};$

here we implicitly assume that the $m_{1}$ random vectors from $K$ and $m_{2}$ random vectors from $\Pi^{\circ}(K)$ are independent. With these notational conventions, we have the following theorem.

As we will explain at the end of § 7, when $m_{1},m_{2}\rightarrow\infty$, Theorem 1.4 also implies Petty’s inequality (1).

For the proof of Theorem 1.4 we first need to establish an empirical version of Minkowski’s first inequality which we believe is of independent interest. In fact, we establish a generalization of the latter, stated as follows.

Theorem 1.4 follows directly from the latter theorem since

$\displaystyle\mathbb{E}$

$\displaystyle V_{1}([K]_{m_{1}},[\Pi^{\circ}(K)]_{m_{2}}^{\infty})=\mathbb{E}V_{1}(\bm{X}_{1}S_{m_{1}},{\bm{X}}_{2}B^{m_{2}}_{\infty}).$

In each of Theorems 1.1, 1.4 and 1.5, we have used a single matrix ${\bm{X}}$ with columns arranged in blocks ${\bm{X}}_{i}$ according to Table 1, and multiple bodies $C_{1},\ldots,C_{n}$. When the $C_{i}$ are all equal to a given compact convex set $C$ in $\mathbb{R}^{m_{1}}$, we have

$$V({\bm{X}}C_{1},\ldots,{\bm{X}}C_{n})=V({\bm{X}}_{1}C,\ldots,{\bm{X}}_{1}C)=\lvert{\bm{X}}_{1}C\rvert,$$

and only the first block ${\bm{X}}_{1}$ is involved in the expression; in particular, the block matrices in the above mixed entry functional are dependent. When the $C_{i}$’s are compact convex sets placed in consecutive orthogonal subspaces $\mathbb{R}^{m_{i}}$, then the use of independent blocks ${\bm{X}}_{i}$ allows for distinct entries in

$$V({\bm{X}}C_{1},\ldots,{\bm{X}}C_{n})=V({\bm{X}}_{1}C_{1},\ldots,{\bm{X}}_{n}C_{n})$$

and all blocks ${\bm{X}}_{i}$ of ${\bm{X}}$ are used (and are independent). The block notation for ${\bm{X}}$ also accommodates scenarios between these two extremes, where some of the $C_{i}$’s are repeated while others are taken in orthogonal subspaces.

We work in Euclidean space $\mathbb{R}^{n}$ and use $|\cdot|$ for the $n$-dimensional Lebesgue measure. The unit Euclidean ball in $\mathbb{R}^{n}$ is $B^{n}_{2}$, while the unit sphere is $S^{n-1}$. We use $P_{H}$ to denote the orthogonal projection onto a subspace $H$. We write $u^{\perp}=\{x\in\mathbb{R}^{n}:\ \langle x,u\rangle=0\}$ for the $(n-1)$-dimensional subspace of $\mathbb{R}^{n}$ that is orthogonal to $u\in S^{n-1}$.

A convex body $K\subset\mathbb{R}^{n}$ is a compact, convex set with non-empty interior. The set of all compact, convex sets is denoted by $\mathcal{K}^{n}$. We say that $K$ is origin-symmetric if $K=-K$. We also say that $K$ is 1-unconditional if $K$ is symmetric with respect to reflections in the standard coordinate hyperplanes.

The support function of $K\in\mathcal{K}^{n}$ is defined by

$$h_{K}(x)=\max\limits_{y\in K}\left<x,y\right>,\quad x\in\mathbb{R}^{n}.$$

The polar body $K^{\circ}$ of an origin-symmetric convex body $K$ in $\mathbb{R}^{n}$ is defined as $K^{\circ}=\{x\in\mathbb{R}^{n}:\ h_{K}(x)\leq 1\}$. The gauge function (or Minkowski functional) of an origin-symmetric convex body $K$ is defined as $||x||_{K}=\inf\{t\geq 0:\ x\in tK\}$. If $K$ contains the origin in its interior, then $||x||_{K}=h_{K^{\circ}}(x)$.

The Minkowski combination of $K_{1},\dots,K_{m}\in\mathcal{K}^{n}$ is defined as

$$\lambda_{1}K_{1}+\dots+\lambda_{m}K_{m}=\{\lambda_{1}x_{1}+\dots+\lambda_{m}x_{m}:\ x_{i}\in K_{i}\}$$

where $\lambda_{1},\dots,\lambda_{m}\geq 0.$ The Minkowski theorem on volume of Minkowski combinations says that

$$|\lambda_{1}K_{1}+\dots+\lambda_{m}K_{m}|=\sum\limits_{i_{1},\dots,i_{n}=1}^{m}\lambda_{i_{1}}\cdots\lambda_{i_{n}}V(K_{i_{1}},\dots,K_{i_{n}}).$$

The coefficient $V(K_{i_{1}},\dots,K_{i_{n}})$ is the mixed volume of $K_{i_{1}},\dots,K_{i_{n}}$; when the last body $K_{i_{n}}$ appears $\ell$ times, we write $V(K_{i_{1}},\ldots,K_{i_{\ell}};K_{i_{n}},\ell)$, where $1\leq\ell\leq n$. For $K,L\in\mathcal{K}^{n}$ and $0\leq i\leq n$, we write $V_{i}(K,L)$ to denote the mixed volume of $K$ repeated $n-i$ times and $L$ repeated $i$ times.

If $K_{1},\dots,K_{n-1}$ are convex bodies in $\mathbb{R}^{n}$ and $u\in S^{n-1}$, then we write $v(P_{u^{\perp}}K_{1},\dots,P_{u^{\perp}}K_{n-1})$ for the $(n-1)$-dimenisonal mixed volume of $P_{u^{\perp}}K_{1},\dots,P_{u^{\perp}}K_{n-1}$ in $u^{\perp}$. It is known (see e.g. [34, p. 302]) that for $u\in S^{n-1}$,

$$v(P_{u^{\perp}}K_{1},\dots,P_{u^{\perp}}K_{n-1})=nV(K_{1},\dots,K_{n-1},[0,u])$$

(8)

where $[0,u]$ denotes the line segment connecting the origin and $u$.

The projection body $\Pi K$ of a convex body $K$ is defined as the origin-symmetric convex body such that $h_{\Pi K}(u)=|P_{u^{\perp}}K|$ for all $u\in S^{n-1}$. It follows from (8) that $h_{\Pi K}(u)=nV_{1}(K,[0,u])$ for $u\in S^{n-1}$. We will denote the polar projection body $(\Pi K)^{\circ}$ by $\Pi^{\circ}K$.

More generally, the mixed projection body $\Pi(K_{1},\dots,K_{n-1})$ of the convex bodies $K_{1},\dots,K_{n-1}$ is defined by

$$h_{\Pi(K_{1},\dots,K_{n-1})}(u)=v(P_{u^{\perp}}K_{1},\dots,P_{u^{\perp}}K_{n-1})=nV(K_{1},\dots,K_{n-1},[0,u])$$

for all $u\in S^{n-1},$ where we used (8) in the last identity. The polar of the mixed projection body $\Pi^{\circ}(K_{1},\dots,K_{n-1})$ is defined as $(\Pi(K_{1},\dots,K_{n-1}))^{\circ}$ so that

$$||\theta||_{\Pi^{\circ}(K_{1},\dots,K_{n-1})}=nV(K_{1},\dots,K_{n-1},[0,\theta]).$$

For background on mixed projection bodies, see [21] and [34].

We recall the notion of $L_{p}$-addition of convex bodies from $L_{p}$-Brunn–Minkowski theory, e.g. [8, 20, 23]. For $K,L\in\mathcal{K}^{n}$ containing the origin and $p\geq 1$, we will write $K+_{p}L$ for their $L_{p}$-sum, i.e.,

$$h^{p}_{K+_{p}L}(u)=h^{p}_{K}(u)+h^{p}_{L}(u),\quad u\in\mathbb{R}^{n}.$$

(9)

A general framework for addition of sets, called $M$-addition, was developed by Gardner, Hug and Weil [10]. Let $M$ be an arbitrary subset of $\mathbb{R}^{m}$ and define the $M$-combination $\oplus_{M}(K_{1},\dots,K_{m})$ of arbitrary sets $K_{1},\dots,K_{m}$ in $\mathbb{R}^{n}$ by

$$\oplus_{M}(K_{1},\dots,K_{m})=\left\{\sum_{i=1}^{m}a_{i}x_{i}:x_{i}\in K_{i},\,(a_{1},\dots,a_{m})\in M\right\}.$$

If $M=\{(1,1)\}$ and $K_{1}$, and $K_{2}$ are convex sets, then $K_{1}\oplus_{M}K_{2}=K_{1}+K_{2}$, i.e., $\oplus_{M}$ is the usual Minkowski addition. If $M=B_{q}^{n},\ q\geq 1$ with $1/p+1/q=1$, and $K_{1}$ and $K_{2}$ are origin-symmetric convex bodies, then $K_{1}\oplus_{M}K_{2}=K_{1}+_{p}K_{2}$, i.e., $\oplus_{M}$ corresponds to $L_{p}$-addition as in (9). More generally, let $\psi:[0,\infty)^{2}\to[0,\infty)$ be convex, increasing in each argument, and $\psi(0,0)=0$, $\psi(1,0)=\psi(0,1)=1$. Let $K$ and $L$ be origin-symmetric convex bodies and let $M=B_{\psi}^{\circ}$, where $B_{\psi}=\{(t_{1},t_{2})\in[-1,1]^{2}:\psi(|t_{1}|,|t_{2}|)\leq 1\}$. Then we define $K+_{\psi}L$ to be $K\oplus_{M}L$. In this way, $M$-addition encompasses previous notions of Orlicz addition, e.g. [25].

It was shown in [10, Section 6] that when $M$ is 1-unconditional and $K_{1},\ldots,K_{m}$ are origin-symmetric and convex, then $\oplus_{M}(K_{1},\dots,K_{m})$ is origin-symmetric and convex. For our purposes, for such $M$ and $C_{1},\ldots,C_{m}$, we have

$$\oplus_{M}({\bf X}_{1}C_{1},\ldots,{\bf X}_{n}C_{n})=[{\bf X}_{1}\cdots{\bf X}_{n}]\oplus_{M}(C_{1},\ldots,C_{n});$$

(10)

see [29, Sections 4 and 5] for further background, details and references.

Let $K\in\mathcal{K}^{n}$ and $u\in S^{n-1}$. We define $f_{K}:P_{u^{\perp}}K\rightarrow\mathbb{R}$ by

$$f_{K}(x)=\sup\{\lambda:x+\lambda u\in K\}$$

and $g_{K}:P_{u^{\perp}}K\rightarrow\mathbb{R}$ by

$$g_{K}(x)=\inf\{\lambda:x+\lambda u\in K\}.$$

Notice that $-f_{K}$ and $g_{K}$ are convex functions.

The Steiner symmetral of a non-empty Borel set $A\subseteq\mathbb{R}^{n}$ of finite measure with respect to $u^{\perp}$, denoted here by $S_{u^{\perp}}A$, is constructed as follows: for each line $l$ orthogonal to $u^{\perp}$ such that $l\cap A$ is non-empty and measurable, the set $l\cap S_{u^{\perp}}A$ is a closed segment with midpoint on $u^{\perp}$ and length equal to the one-dimensional measure of $l\cap A$. In particular, if $K$ is a convex body

$$S_{u^{\perp}}K=\left\{x+\lambda u:x\in P_{u^{\perp}}K,-\dfrac{f_{K}(x)-g_{K}(x)}{2}\leq\lambda\leq\dfrac{f_{K}(x)-g_{K}(x)}{2}\right\}.$$

This shows that $S_{u^{\perp}}K$ is convex, since the function $f_{K}-g_{K}$ is concave. Moreover, $S_{u^{\perp}}K$ is symmetric with respect to $u^{\perp}$, it is closed, and by Fubini’s theorem it has the same volume as $K$.

For a Borel set $A\subset\mathbb{R}^{n}$ with finite volume, the symmetric rearrangement $A^{*}$ of $A$ is the open Euclidean ball centered at the origin whose volume is equal to the volume of $A$. The symmetric decreasing rearrangement of $\mathds{1}_{A}$ is defined as $\mathds{1}_{A}^{*}=\mathds{1}_{A^{*}}$. It will be convenient to use the following bracket notation for indicator functions:

$$\mathds{1}_{A}(x)=\left\llbracket x\in A\right\rrbracket.$$

(11)

Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}$ be an integrable function. Its layer-cake representation is given by

$$f(x)=\int_{0}^{\infty}\left\llbracket x\in\{f>t\}\right\rrbracket dt.$$

(12)

The symmetric decreasing rearrangement $f^{*}$ of $f$ is defined by

$$f^{*}(x)=\int\limits_{0}^{\infty}\left\llbracket x\in\{f>t\}^{*}\right\rrbracket dt.$$

The function $f^{*}$ is radially-symmetric, radially decreasing and equimeasurable with $f$, i.e. $\{f>t\}$ and $\{f^{*}>t\}$ have the same volume for each $t>0.$ For integrable functions $f$, the Steiner symmetral $f^{*}(\cdot|u)$ of $f$ with respect to $u^{\perp}$ is defined as follows:

$\displaystyle f^{*}(x|u)=\int_{0}^{\infty}\left\llbracket x\in S_{u^{\perp}}\{f>t\}\right\rrbracket$

In other words, we obtain $f^{*}(\cdot|u)$ by rearranging $f$ along every line parallel to $u$. For more background on rearrangements, see [17].

Rogers–Shephard [32] and Shephard [36] systematized the use of Steiner symmetrization as a means of proving geometric inequalities with their introduction of linear parameter systems and shadow systems, respectively. A linear parameter system is a family of sets

$$K_{t}=\mathop{\rm conv}\{x_{i}+\alpha_{i}tu:i\in I\},$$

(13)

where $\{\alpha_{i}\}_{i\in I}$ and $\{x_{i}\}_{i\in I}$ are bounded sets, and $I$ is an index set. For a unit vector $u\in\mathbb{R}^{n}$ and a convex body $\mathcal{C}$ in $\mathbb{R}^{n+1}=\mathbb{R}^{n}\oplus\mathbb{R}$, a shadow system is a family of sets of the form

$$K_{t}=P_{t}\mathcal{C},$$

where $P_{t}:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n}$ is the projection parallel to $e_{n+1}-tu$. Setting

$$\mathcal{C}=\mathop{\rm conv}\{x+\alpha(x)e_{n+1}:x\in K\subseteq e_{n+1}^{\perp}\}$$

where $\alpha$ is a bounded function on $K$, gives rise to the shadow system for $t\in[0,1]$,

$$K_{t}=\mathop{\rm conv}\{x+t\alpha(x):x\in K\}.$$

The choice of $\alpha(x)=-g(P_{u^{\perp}}x)-f(P_{u^{\perp}}x)$ has the property that $K_{0}=K$, while $K_{1/2}$ is the Steiner symmetral of $K$ about $u^{\perp}$, and $K_{1}$ is the reflection of $K$ about $u^{\perp}$. For background on linear parameter systems and shadow systems, we refer to [34, 4].

We will make essential use of the following fundamental theorem of Shephard

A non-negative, non-identically zero function $f$ is called log-concave if $\log f$ is concave on $\{f>0\}$. We note that if $f$ is a convex function, then the function $\left\llbracket f(x)\leq 1\right\rrbracket$ is $\log$-concave. Also, $f$ is quasi-concave if for all $t$ the set $\{x:f(x)>t\}$ is convex, and $f$ is quasi-convex if for all $t$ the set $\{x:f(x)\leq t\}$ is convex.

The Prékopa–Leindler inequality states that for $0<\lambda<1$ and functions $f,g,h:\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}$ such that for any $x,y\in\mathbb{R}^{n}$

$$h(\lambda x+(1-\lambda)y)\geq f(x)^{\lambda}g(y)^{1-\lambda},$$

the following inequality holds

$$\int_{\mathbb{R}^{n}}h\geq\left(\int_{\mathbb{R}^{n}}f\right)^{\lambda}\left(\int_{\mathbb{R}^{n}}g\right)^{1-\lambda}.$$

We will use the following consequence of the Prékopa–Leindler inequality: if $f:\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}_{+}$ is $\log$-concave, then

$$g(x)=\int_{\mathbb{R}^{m}}f(x,y)dx$$

is a $\log$-concave function on $\mathbb{R}^{n}$; see [13].

We also make use of Christ’s form [5] of the Rogers–Brascamp–Lieb–Lutinger inequality, see the survey [29] for the related inequalities, their applications and further references.

When each $F_{u,\omega}$ is even and quasi-convex, then the inequality in Theorem 3.2 is reversed.

For subsequent reference, we note that the theorem is proved by iterated Steiner symmetrization and the key step involves Steiner symmetrization as follows:

	$\displaystyle\int_{\mathbb{R}^{n}}\cdots\int_{\mathbb{R}^{n}}F(x_{1},\dots,x_{m})\prod_{i=1}^{m}f_{i}(x_{i})dx_{1}\dots dx_{m}=$
	$\displaystyle=\int_{(u^{\perp})^{m}}\int_{\mathbb{R}^{m}}F_{u,\omega}(t_{1},\dots,t_{m})\prod_{i=1}^{m}f_{i}(\omega_{i}+t_{i}u)dt_{1}\dots dt_{m}d\omega_{1}\dots d\omega_{m}$
	$\displaystyle\leq\int_{(u^{\perp})^{m}}\int_{\mathbb{R}^{m}}F_{u,\omega}(t_{1},\dots,t_{m})\prod_{i=1}^{m}f^{*}_{i}(\omega_{i}+t_{i}u|u)dt_{1}\dots dt_{m}d\omega_{1}\dots d\omega_{m}$
	$\displaystyle=\int_{\mathbb{R}^{n}}\cdots\int_{\mathbb{R}^{n}}F(x_{1},\dots,x_{m})\prod_{i=1}^{m}f^{*}_{i}(x_{i}|u)dx_{1}\dots dx_{m},$

where $f_{i}^{*}(\cdot|u)$ is the Steiner symmetral of $f_{i}$ in direction $u$.