Weighted Berwald’s Inequality

Let $\mathbb{R}^{n}$ be the standard $n$-dimensional real vector space with the Euclidean structure. We write $\text{\rm Vol}_{m}(C)$ for the $m$-dimensional Lebesgue measure (volume) of a measurable set $C\subset\mathbb{R}^{n}$, where $m=1,...,n$ is the dimension of the minimal affine space containing $C$. The volume of the unit ball $B_{2}^{n}$ is written as $\kappa_{n},$ and its boundary, the unit sphere, will be denoted as usual $\mathbb{S}^{n-1}.$ A set $K\subset\mathbb{R}^{n}$ is said to be convex if for every $x,y\in K$ and $\lambda\in[0,1],$ $(1-\lambda)x+\lambda y\in K.$ We say $K$ is a convex body if it is a convex, compact set with non-empty interior; the set of all convex bodies in $\mathbb{R}^{n}$ will be denoted by $\mathcal{K}^{n}$. The set of those convex bodies containing the origin will be denoted $\mathcal{K}^{n}_{0}.$ A convex body $K$ is centrally symmetric, or just symmetric, if $K=-K$. There exists an addition on the set of convex bodies: the Minkowski sum of $K$ and $L$, and one has that $K+L=\{a+b:a\in K,b\in L\}.$

We recall that a non-negative function $f$ is said to be concave on $\mathbb{R}^{n}$ if for every $x,y\in\mathbb{R}^{n}$ and $\lambda\in[0,1]$ one has

and that the support of a function is precisely $\text{supp}(f)=\overline{\{x\in\mathbb{R}^{n}:f(x)>0\}}.$ One can see that a non-negative, concave function will be supported on a convex set. It is easy to show if a non-negative, concave function takes the value infinity anywhere on its support, then the function is identically infinity on the interior of its support from convexity; therefore, throughout this paper, given a non-negative, concave function $f,$ we shall assume it is not identically infinity, and so $f$ will have a finite maximum value, denoted $\|f\|_{\infty}$. If $K$ is the support of a non-negative, concave function $f$, then $K_{t}=\{x\in\mathbb{R}^{n}:f(x)\geq t\}=\{f\geq t\}$ are the level sets of $f$. Notice that the level sets are also convex. Additionally, if $\|f\|_{\infty}=f(0),$ then $0\in K_{t}$ for all $t\leq\|f\|_{\infty}.$ If $f$ is even, then $K$ is symmetric and so too is each $K_{t}.$ In any case, if $K$ is also bounded, then each $K,K_{t}\in\mathcal{K}^{n}$ (for each $t\leq\|f\|_{\infty}$).

We next recall that the classical Berwald inequality states that if $f$ is a non-negative, concave function supported on some convex set $K\subset\mathbb{R}^{n},$ then, the function given by

is decreasing for $p\in(-1,\infty)$ [4] with equality [22] if and only if the graph of $f$ is a certain cone with $K$ as its base. Here, the combinatorial coefficients are given by $\binom{m}{p}=\frac{\Gamma(m+1)}{\Gamma(p+1)\Gamma(m-p+1)},$ with $\Gamma(z)$ the standard Gamma function, defined for $z\in\mathbb{C}$ except for when $z$ is negative integer. Usually written in the form $t_{f}(q)\leq t_{f}(p)$ for $-1<p\leq q<\infty,$ Berwald’s inequality has several applications in the fields of convex geometry and probability theory, see for example [22, 39, 5, 24]. The first goal of this paper is to establish generalizations of Berwald’s inequality for measures with density under certain concavity assumptions. We will also analyze equality conditions; in particular, we obtain equality conditions for the classical Berwald inequality by a method independent of other proofs (e.g., [1, 22, 9]). To accomplish these tasks, we first prove a generalized Berwald’s inequality, Lemma 2.1.

We will say a Borel measure $\mu$ has density if it has a locally integrable Radon-Nikodym derivative from $\mathbb{R}^{n}$ to $\mathbb{R}^{+}$, i.e,

A Borel measure $\mu$ on $\mathbb{R}^{n}$ is said to be $F$-concave on a class $\mathcal{C}$ of compact subsets of $\mathbb{R}^{n}$ if there exists a continuous, (strictly) monotonic, invertible function $F:(0,\mu(\mathbb{R}^{n}))\to(-\infty,\infty)$ such that, for every pair $A,B\in\mathcal{C}$ and every $t\in[0,1]$, one has

When $F(x)=x^{s},s>0$ this can be written as

and we say $\mu$ is $s$-concave. When $s=1$, we merely say the measure is concave. In the limit as $s\rightarrow 0$, we obtain the case of log-concavity, which can also be obtained by taking $F(x)=\log x$:

The classical Brunn-Minkowski inequality (see for example [21]) asserts the $1/n$-concavity of the Lebesgue measure on the class of all compact subsets of $\mathbb{R}^{n}$. From Borell’s classification on concave measures [7], a Radon measure (locally finite and regular Borel measure) is log-concave on Borel subsets of $\mathbb{R}^{n}$ if, and only if, $\mu$ has a density $\phi(x)$ that is log-concave, i.e. $\phi(x)=Ae^{-\psi(x)},$ where $A>0$ and $\psi:\mathbb{R}^{n}\to\mathbb{R}^{+}$ is convex. Similarly, a Radon measure is $s$-concave on Borel subsets of $\mathbb{R}^{n}$, $s\in(-\infty,0)\cup(0,1/n),$ if, and only if, $\mu$ has a density $\phi(x)$ that is $p$-concave (if $s>0$) or $p$-convex (if $s<0$), where $p=s/(1-ns).$ However, all we will require is that a measure is $s$-concave on a class of convex sets; we will discuss an important example below. Thus, our results in the case of $s$-concave measures include measures beyond Borell’s classification.

We can now state our first main result, which is the Berwald inequality for $F$-concave measures under different restrictions on the function $F$. This result applies to a variety of measures, including $s$-concave ones.

We obtain the following corollary for $s$-concave measures; the case where $s<0$ was previously done by Fradelizi, Guédon and Pajor [19], by modifying Borell’s proof [8] of the classical inequality of Berwald. Presented in [20] is a proof for all $s\in\mathbb{R},$ based on techniques from a work by Koldobsky, Pajor and Yaskin [27]. Both extensions do not mention equality conditions.

The equality conditions to Corollary 1.2 may seem a bit strange; we are able to obtain an exact formula for the function $f$ when the measure $\mu$ is $s$-concave and $1/s$-homogeneous, $s\in(0,1/n]$. Recall that a Borel measure $\mu$ is said to be $\alpha$-homogeneous for some $\alpha>0$ if $\mu(tA)=t^{\alpha}\mu(A)$ for all compact sets $A\subset\operatorname{supp}\mu$ and $t>0$ so that $tA\subset\operatorname{supp}\mu.$ If $\mu$ has density $\phi$, then one can check using the Lebesgue differentiation theorem that this implies that $\phi$ is $(\alpha-n)-$homogeneous.

We say a set $L$ with $0\in\text{int}(L)$ is star-shaped if every line passing through the origin crosses the boundary of $L$ exactly twice. We say $L$ is a star body if it is a compact, star-shaped set whose radial function $\rho_{L}:\mathbb{R}^{n}\setminus\{0\}\to\mathbb{R},$ given by $\rho_{L}(y)=\sup\{\lambda:\lambda y\in L\},$ is continuous. For $K\in\mathcal{K}^{n}_{0},$ the Minkowski functional of $K$ is defined to be $\|y\|_{K}=\rho^{-1}_{K}(y)=\inf\{r>0:y\in rK\}.$ The Minkowski functional $\|\cdot\|_{K}$ of $K\in\mathcal{K}^{n}_{0}$ is a norm on $\mathbb{R}^{n}$ if $K$ is symmetric. If $x\in\mathbb{R}^{n}$ and $L\subset\mathbb{R}^{n}$ satisfy that $L-x$ is a star body, then the generalized radial function of $L$ at $x$ is defined by $\rho_{L}(x,y):=\rho_{L-x}(-y)$. Note that for every $K\in\mathcal{K}^{n},$ $K-x$ is a star body for every $x\in\text{int}(K)$.

One gets the following formula for $\mu(K)$ when $\mu$ is $\alpha$-homogeneous, $\alpha>0$, and $K$ is a star body in $\mathbb{R}^{n}$.

Crucial to the statement of equality conditions, and our investigations henceforth, will be the roof function associated to a star body $K$, which we define as $\ell_{K}(0)=1,\ell_{K}(x)=0$ for $x\neq K$ and, for $x\in K\setminus\{0\},$ $\ell_{K}(x)=\left(1-\frac{1}{\rho_{K}(x)}\right).$ In polar coordinates, $\ell_{K}(r\theta)$ becomes an affine function in $r$ for $r\in[0,\rho_{K}(\theta)]$:

Note that if $K\in\mathcal{K}^{n}_{0},$ then we can also write $\ell_{K}(x)=1-\|x\|_{K}$ for $x\in K$ and $0$ otherwise. Observe that, for a non-negative, concave function supported on some $K\in\mathcal{K}^{n}_{0}$ one obtains for $\theta\in\mathbb{S}^{n-1}$ and $r\in[0,\rho_{K}(\theta)]$ that

we will make liberal use of this bound throughout this work. Functions of the form $f(x)=M\ell_{K-x_{0}}(x-x_{0})$ for some $M>0$ and $x_{0}\in K$ will also be referred to roof functions, with height $M$ and vertex $x_{0}$. The reason for this vocabulary will become clearer below.

Using (2), one can verify by hand that the function $\ell_{K}(x)$ satisfies, for $\mu$ an $s$-concave, $1/s$-homogeneous measure, that

Therefore, $\ell_{K}(x)$ yields equality in the Berwald inequality for $s$-concave measures, Corollary 1.2, under the additional assumption that $\mu$ is $1/s$-homogeneous. The next theorem shows this is the only such function.

Gardner and Zhang [22] gave a different proof, which yields that equality is satisfied in the classical Berwald inequality precisely when the graph of $f$ is a certain cone with $K$ as a base, i.e. that $f$ is a roof function. In Corollary 1.4, we obtain a proof using Theorem 1.3, verifying that our techniques reduce to the known result. We must also mention that this result was also obtained in [1, Theorem 7.2] via a different technique. In that work, the roof function was defined via its graph in $\mathbb{R}^{n+1}.$ Specifically they constructed the roof function in the following way: given a convex set $K\subset\mathbb{R}^{n}$ (which will become the base of a hypercone), let $M>0$ be the height of the hypercone, and let $x_{0}\in K$ be the location of the projection of vertex of the hypercone. Then, the roof function with height $M$ and vertex $x_{0}$ is equivalently defined as the non-negative, concave function $f$ whose graph is given by

where conv denotes the convex hull. From this formulation, we obtain an interesting formula for the level sets of a roof function $f:$ for $0\leq t\leq M,$ one has that $K_{t}=\frac{t}{M}x_{0}+(1-\frac{t}{M})K.$

We next list two applications for the standard Gaussian measure on $\mathbb{R}^{n},$ which we recall is given by $d\gamma_{n}(x)=\frac{1}{(2\pi)^{n/2}}e^{-|x|^{2}/2}dx.$ From Borell’s classification, we see that the Gaussian measure is log-concave on $\mathbb{R}^{n}$ over any collection of compact sets closed under Minkowski summation. Thus, we can apply the second case of Corollary 1.2 and obtain a Berwald-type inequality for the Gaussian measure in this case. However, the Ehrhard inequality shows one can improve on the log-concavity of the Gaussian measure: For $0<t<1$ and Borel sets $K$ and $L$ in $\mathbb{R}^{n}$, we have

i.e. $\Phi^{-1}\circ\gamma_{n}$ is concave, where $\Phi(x)=\gamma_{1}((-\infty,x))$. The inequality (5) was first proven by Ehrhard for the case of two closed, convex sets [15, 14]. Latała [31] generalized Ehrhard’s result to the case of an arbitrary Borel set $K$ and convex set $L$; the general case for two Borel sets of the Ehrhard’s inequality was proven by Borell [10]. Since $\Phi$ is log-concave, the log-concavity of the Gaussian measure is strictly weaker than the Ehrhard inequality. Additionally, Kolesnikov and Livshyts showed that the Gaussian measure is $\frac{1}{2n}$ concave on $\mathcal{K}^{n}_{0},$ the set of convex bodies containing the origin in their interior [28]. That is, by restricting the admissible sets in the concavity equation, the concavity can improve.

The equality condition for the third case of Corollary 1.5 can be deduced from Theorem 1.1, so we do not explicitly state it. If one further restricts the admissible sets, one can do even better. The Gardner-Zvavitch inequality states for symmetric $K,L\in\mathcal{K}^{n}_{0}$ and $t\in[0,1]$ that

i.e. $\gamma_{n}$ is $1/n$-concave over the class of symmetric convex bodies. This inequality was first conjectured in [23] by Gardner and Zvavitch; a counterexample was shown in [40] when $K$ and $L$ are not symmetric. Important progress was made in [28], which lead to the proof of the inequality (6) by Eskenazis and Moschidis in [16] for symmetric convex bodies. Recently, Cordero-Erausquin and Rotem [12] extended this result to the class

That is, every measure $\mu\in\mathcal{M}_{n}$ is $1/n$-concave over the class of symmetric convex bodies. To show how rich this class is, $\mathcal{M}_{n}$ includes not only every rotationally invariant, log-concave measure (e.g. Gaussian), but also Cauchy-type measures. Combining these results, we obtain a Berwald-type inequality.

We emphasize that the $(1/2n)$-concavity of the Gaussian measure on $\mathcal{K}^{n}_{0}$ shown in [28] and the $1/n$-concavity of $\gamma_{n}$ and other measures from $\mathcal{M}_{n}$ over the class of symmetric convex bodies falls strictly outside the classification of $s$-concave measures by Borell. This paper is organized as follows. In Section 2, we prove a version of Berwald’s inequality for $F$-concave measures. In Section 3, we discuss surface area measure, projection bodies, and radial mean bodies. Then, we apply our results to weighted generalizations of radial mean bodies. Along the way, we obtain more inequalities of Rogers and Shephard and of Zhang type. We would like to mention here that weighted extensions of concepts from the Brunn-Minkowski theory is a very rich field. This includes works on the surface area measure [3, 41, 33, 32, 36] and general measure extensions of the projection body of a convex body [35, 30]. Recently, it has been shown that these developments, in particular the concavities for the Gaussian measure and Borell’s classification, have led to a burgeoning weighted Brunn-Minkowski theory, see [29, 17, 18].

In this section, we establish a generalization of Berwald’s inequality. In what follows, for a finite Borel measure $\mu$ and a Borel set $K$ with positive $\mu$-measure, $\mu_{K}$ will denote the normalized probability on $K$ with respect to $\mu$, that is for measurable $A\subset\mathbb{R}^{n}:$ $\mu_{K}(A)=\frac{\mu(K\cap A)}{\mu(K)}.$ Notice that for every non-negative, measurable function $f$ on $K$ and $p>0$ such that $f\in L^{p}(\mu,K)$, one has the layer cake formula

from the following use of Fubini’s theorem:

Additionally, if $\mu$ is $F$-concave, with $F$ increasing and invertible, on a class $\mathcal{C}$ of convex sets, then for $K\in\mathcal{C}$ in the support of a concave function $f$, one has that the function given by $f_{\mu}(t)=\mu_{K}(\left\{f\geq t\right\})$ is $\tilde{F}$-concave, where $\tilde{F}(x)=F(\mu(K)x),$ as long as the level sets of $f$ belong to $\mathcal{C}.$ Indeed, since $f$ is concave, one has, for $\lambda\in[0,1]$ and $u,v\geq 0$, that

Using the $F$-concavity of $\mu,$ this yields

Inserting the definition of $\tilde{F}$ and $f_{\mu},$ this is precisely

We next need the appropriate layer cake formula for when $p<0.$ Notice that for every non-negative, measurable function $f$ on a Borel set $K$ and $p<0$ such that $f\in L^{p}(\mu,K)$ for a Borel measure $\mu$, one has the layer cake formula

from the following use of Fubini’s theorem: