On Extensions of the Loomis-Whitney Inequality and Ball's Inequality for Concave, Homogeneous Measures

TL;DR

This paper extends the Loomis-Whitney and Ball inequalities to a broader class of measures that are q-concave and 1/q-homogeneous, broadening their applicability in convex geometry.

Contribution

It generalizes classical inequalities to q-concave, 1/q-homogeneous measures, providing new bounds for convex bodies under these measures.

Findings

Extended Loomis-Whitney inequality to q-concave measures

Generalized Ball's inequality for new measure classes

Provided bounds applicable to a wider range of convex bodies

Abstract

The Loomis-Whitney inequality states that the volume of a convex body is bounded by the product of volumes of its projections onto orthogonal hyperplanes. We provide an extension of both this fact and a generalization of this fact due to Ball to the context of $q-$concave, $\frac{1}{q}-$homogeneous measures.

Equations

$$|K{|}^{n-1}$$

$$\sum _{i=1}^m{c}_i{u}_i\otimes u_i=I_n,$$

$$|K{|}^{n-1}$$

$$\sum _{i=1}^m{c}_i=n,$$

$$f(\lambda x+(1-\lambda )y)$$

$$\mu (\lambda E+(1-\lambda )F)$$

$${\mu }_1(A,B)=\underset{\epsilon \to 0}{\mathrm{liminf}}\frac{\mu (A+\epsilon B)-\mu (A)}{\epsilon }$$

$${\mu }_1(K,E+tF)={\mu }_1(K,E)+t{\mu }_1(K,F)$$

$$\mu (A{)}^{1-q}\mu (B{)}^q$$

$$P_{\mu ,K}(\theta )=\frac{n}{2}{\int }_0^1{\mu }_1(tK,[-\theta ,\theta ])dt$$

$$\mu (K{)}^{n+\frac{1}{p}-1}\le 2^{n+\frac{1}{p}}{\left(1+\frac{1}{pn}\right)}^n{\left(\sum _{k=1}^n{\tilde{g}}^p(u_k)\right)}^{-\frac{1}{p}}\prod _{i=1}^n{P}_{\mu ,K}(u_i{)}^{1+\frac{{\tilde{g}}^p(u_i)}{p{\sum }_{k=1}^n{\tilde{g}}^p(u_k)}}.$$

$$\mathcal{P}=\mathcal{P}((u_i{)}_{i=1}^m):=\mathcal{S}\cup {\mathcal{S}}^{(1)}\cup ...\cup {\mathcal{S}}^{(n-1)},$$

$$\sum _{i=1}^m{c}_i{u}_i\otimes u_i=I_n$$

$$\mu (K{)}^{n+\frac{1}{p}-1}$$

$$F_i=\{u={\alpha }_i{u}_i+\sum _{j\ne i}{\beta }_j{u}_j:|{\beta }_j|\le {\alpha }_j\},$$

$${\mu }_{n-1}(F_i)\ge {\left(\frac{pn}{pn+1}\right)}^n\left(1+\frac{{\tilde{g}}^p(u_i)}{p{\sum }_{k=1}^n{\tilde{g}}^p(u_k)}\right){\left(\sum _{i=1}^n{\tilde{g}}^p(u_i)\right)}^{\frac{1}{p}}{\alpha }_i^{-1}\prod _{j=1}^n{\alpha }_j^{1+\frac{{\tilde{g}}^p(u_j)}{p{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}},$$

$${\mu }_{n-1}(F_1)$$

$$=\sum _{\sigma =(\pm 1,...,\pm 1)}{\int }_0^{{\alpha }_n}...{\int }_0^{{\alpha }_2}g\left({\alpha }_1{u}_1+\sum _{j=2}^n{\beta }_j\sigma (j)u_j\right)d{\beta }_2...d{\beta }_n$$

$$=\sum _{\sigma =(\pm 1,...,\pm 1)}{\int }_0^{{\alpha }_n}...{\int }_0^{{\alpha }_2}{\left({\alpha }_1+\sum _{j=2}^n{\beta }_j\right)}^{\frac{1}{p}}g\left(\frac{{\alpha }_1}{{\alpha }_1+{\sum }_{j=2}^n{\beta }_j}{u}_1+\sum _{j=2}^n\frac{{\beta }_j}{{\alpha }_1+{\sum }_{j=2}^n{\beta }_j}\sigma (j)u_j\right)d{\beta }_2...d{\beta }_n$$

$$=\sum _{\sigma =(\pm 1,...,\pm 1)}{I}_{\sigma }.$$

$${\mu }_{n-1}(F_1)$$

$$I_{{\sigma }^{\prime }}$$

$$={\int }_0^{{\alpha }_n}...{\int }_0^{{\alpha }_2}{\left({\alpha }_1{\tilde{g}}^p(u_1)+\sum _{j=2}^n{\beta }_j{\tilde{g}}^p(u_j)\right)}^{\frac{1}{p}}d{\beta }_2...d{\beta }_n$$

$$={\left(\sum _{i=1}^n{\tilde{g}}^p(u_i)\right)}^{\frac{1}{p}}{\int }_0^{{\alpha }_n}...{\int }_0^{{\alpha }_2}{\left({\alpha }_1\frac{{\tilde{g}}^p(u_1)}{{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}+\sum _{j=2}^n{\beta }_j\frac{{\tilde{g}}^p(u_j)}{{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}\right)}^{\frac{1}{p}}d{\beta }_2...d{\beta }_n.$$

$${\alpha }_1\frac{{\tilde{g}}^p(u_1)}{{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}+\sum _{j=2}^n{\beta }_j\frac{{\tilde{g}}^p(u_j)}{{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}$$

$$I_{{\sigma }^{\prime }}$$

$$=\left(1+\frac{{\tilde{g}}^p(u_1)}{p{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}\right){\left(\sum _{i=1}^n{\tilde{g}}^p(u_i)\right)}^{\frac{1}{p}}{\alpha }_1^{-1}\prod _{j=1}^n\frac{1}{1+\frac{{\tilde{g}}^p(u_j)}{p{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}}{\alpha }_j^{1+\frac{{\tilde{g}}^p(u_j)}{p{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}}.$$

$$\prod _{j=1}^n\left(1+\frac{{\tilde{g}}^p(u_j)}{p{\sum }_{i=1}^n{\tilde{g}}^p(u_i)}\right)$$

$$I_{{\sigma }^{\prime }}$$

$$Z=\sum _{i=1}^m[-x_i,x_i].$$

$${\mu }_1(K,Z)=\sum _{i=1}^m{\alpha }_i{\mu }_1(K,[-u_i,u_i])$$

$$P_{\mu ,K}(u_i)$$

$$=\frac{n}{2}{\int }_0^1{t}^{\frac{1}{q}-1}dt{\mu }_1(K,[-u_i,u_i])$$

$$=\frac{qn}{2}{\mu }_1(K,[-u_i,u_i])$$

Full text

1. Introduction

The Loomis-Whitney inequality [LW49] is a well-known geometric inequality concerning convex bodies, compact and convex sets with nonempty interior. Explicitly, the inequality states that if $u_{1},...,u_{n}$ form an orthonormal basis of $\mathbb{R}^{n}$ and $K$ is a convex body in $\mathbb{R}^{n}$, then

[TABLE]

where $K|u_{i}^{\perp}$ denotes the projection of $K$ onto $u_{i}^{\perp}$, the hyperplane orthogonal to $u_{i}$. Equality occurs if and only if $K$ is a box with faces parallel to the hyperplanes $u_{i}^{\perp}$. This was generalized by Ball [Bal91], who showed that if $u_{1},...,u_{m}$ are vectors in $\mathbb{R}^{n}$ and $c_{1},...,c_{m}$ positive constants such that

[TABLE]

then

[TABLE]

Here $u_{i}\otimes u_{i}$ denotes the rank $1$ projection onto the span of $u_{i}$, so $(u_{i}\otimes u_{i})(x)=\langle x,u_{i}\rangle u_{i}$ with $\langle\cdot,\cdot\rangle$ representing the standard Euclidean inner product, and $I_{n}$ is the identity on $\mathbb{R}^{n}$. What will be useful later is the fact that

[TABLE]

which follows by comparing traces in (1.1).

The Loomis-Whitney inequality and Ball’s inequality have been the subject of various generalizations. For instance, Huang and Li [HL17] provided an extension of Ball’s inequality with intrinsic volumes replacing volumes and an arbitrary even isotropic measure replacing the discrete measure $\sum_{i=1}^{m}c_{i}\delta_{u_{i}}$ in the condition $\int_{S^{n-1}}u\otimes u\ d\left(\sum_{i=1}^{m}c_{i}\delta_{u_{i}}\right)(u)=I_{n}$ of (1.1). They [LH16] also demonstrated the $L_{p}$ Loomis-Whitney inequality for even isotropic measures, while Lv [Lv19] very recently demonstrated the $L_{\infty}$ Loomis-Whitney inequality.

In this paper, we will first give a generalization of the original Loomis-Whitney inequality to the context of $q-$concave, $\frac{1}{q}-$homogeneous measures. Using a different argument, we shall then prove a generalization of Ball’s inequality. Our two theorems are independent in the sense that the first is not recovered when specializing the second to the case of $u_{1},...u_{n}$ being an orthonormal basis and $c_{1}=...=c_{n}=1$. Therefore, in fact, two different extensions of the Loomis-Whitney inequality are given.

Let us recall the necessary definitions.

Definition 1.1**.**

A function $f:\mathbb{R}^{n}\to[0,\infty]$ is $p-$concave for some $p\in\mathbb{R}\setminus\{0\}$ if for all $\lambda\in[0,1]$ and $x,y\in\text{supp}(f)$ we have

[TABLE]

Definition 1.2**.**

A function $f:\mathbb{R}^{n}\to[0,\infty]$ is $r-$homogeneous if for all $a>0,x\in\mathbb{R}^{n}$ we have $f(ax)=a^{r}f(x)$.

We will interested in the functions $g$ that are both $s-$concave for some $s>0$ and $\frac{1}{p}-$homogeneous for some $p>0$. In this case, we get that in fact $g$ is $p$-concave (see e.g. Livshyts [Liv]). Continuity will be assumed throughout. An example of a $p-$concave, $\frac{1}{p}-$homogeneous function is $g(x)=1_{\langle x,\theta\rangle>0}\langle x,\theta\rangle^{\frac{1}{p}}$, where $\theta$ is a vector. All such functions $g$, with the exception of constant functions, will be supported on convex cones. To see this, observe that concavity implies that the support is convex and homogeneity implies that if $x\in\text{supp}(g)$ then $tx\in\text{supp}(g)$ for all $t>0$. Moreover, we cannot have both $x,-x\in\text{supp}(g)$, for then concavity will give $g(0)=g\left(\frac{1}{2}x+\frac{1}{2}(-x)\right)>0$, but $g(0)=0$ by homogeneity.

A notation we will use is $\tilde{g}(x)=g(x)+g(-x).$

If $\mu$ is a measure with a $p-$concave, $\frac{1}{p}-$homogeneous density, then a change of variables will show that $\mu$ is $n+\frac{1}{p}$ homogeneous, that is $\mu(tK)=t^{n+\frac{1}{p}}\mu(K)$. From a result of Borell [Bor75], we also have concavity:

Lemma 1.3** (Borell).**

Let $p\in\left(-\frac{1}{n},\infty\right]$ and let $\mu$ be a measure on $\mathbb{R}^{n}$ with $p-$concave density $g$. For $q=\frac{1}{n+\frac{1}{p}}$, $\mu$ is a $q-$concave measure, that is for measurable sets $E,F$ and $\lambda\in[0,1]$ we have

[TABLE]

To now define the generalized notion of projection for measures, one requires the definition of mixed measures (see e.g. Livshyts [Liv]).

Definition 1.4**.**

Let $A,B$ be measurable sets in $\mathbb{R}^{n}$. We define

[TABLE]

to be the mixed $\mu-$measure of $A$ and $B$.

An important simple fact, which follows from Lemma 3.3 in Livshyts [Liv], is that mixed measure is linear in the second variable, so

[TABLE]

for $t\geq 0$.

For $q-$concave measures, we have the following generalization of Minkowski’s first inequality (see e.g. Milman and Rotem [MR14]):

Lemma 1.5**.**

Let $\mu$ be a $q-$concave measure and $A,B$ be measurable sets in $\mathbb{R}^{n}$. Then,

[TABLE]

We now turn to discussing the generalized notion of projection. This notion, defined by Livshyts [Liv], is

[TABLE]

for $\theta\in S^{n-1}$, where $K$ is a convex body, $\mu$ is an absolutely continuous measure, and $[-\theta,\theta]=\{t\theta:t\in[-1,1]\}$. This is a natural extension of the identity $|K|\theta^{\perp}|=\frac{1}{2}\lambda_{1}(K,[-\theta,\theta])$, with $\lambda$ denoting Lebesgue measure, which can be readily seen for polytopes and follows in the general case by approximation.

In [Liv], a version of the Shephard problem for $q-$concave, $\frac{1}{q}-$homogeneous measures was proven with this notion of measure. The author in [Hos] studied the related section and projection comparison problems, including for this same class of $q-$concave, $\frac{1}{q}-$homogeneous measures.

With (1.4), we can now state our first theorem:

Theorem 1.6**.**

Let $\mu$ be a measure with $p-$concave, $\frac{1}{p}-$homogeneous density $g$ for some $p>0$. Then, for any convex body $K$ and an orthonormal basis $(u_{i})_{i=1}^{n}$ with $[-u_{i},u_{i}]\cap\text{supp}(g)\neq\varnothing$ for each $1\leq i\leq n$,

[TABLE]

Before we state our generalization of Ball’s inequality, we introduce another definition. Let $\mathcal{S}=\{(u_{i})_{i=1}^{m}\}$ be a set of unit vectors in $\mathbb{R}^{n}$. Then we define $\mathcal{S}^{(1)}$ to be the set of $u_{ij}=\frac{u_{i}-\langle u_{i},u_{j}\rangle u_{j}}{|u_{i}-\langle u_{i},u_{j}\rangle u_{j}|},$ the normalized projection of $u_{i}$ onto the hyperplane $u_{j}^{\perp}$, for $1\leq i,j\leq m.$ Recursively defining $S^{(k)}=(S^{(k-1)})^{(1)}$, we set

[TABLE]

some finite sets depending on our initial choice of $\{(u_{i})_{i=1}^{m}\}$. Our generalization of Ball’s inequality is the following:

Theorem 1.7**.**

Let $\mu$ be a measure with $p-$concave, $\frac{1}{p}-$homogeneous density $g$ for some $p>0$. If $(u_{i})_{i=1}^{m}$ are unit vectors in $\mathbb{R}^{n}$ and $(c_{i})_{i=1}^{m}$ are positive constant such that

[TABLE]

and moreover $[-u,u]\cap\text{supp}(g)\neq\varnothing$ for each $u\in\mathcal{P}((u_{i})_{i=1}^{m})$, then

[TABLE]

for any convex body $K$.

Observe that the condition $[-u,u]\cap\text{supp}(g)\neq\varnothing$ is not particularly restrictive. For instance, if we consider $g$ whose support is a half space with boundary a half plane $P$, then the condition simply reduces to the fact that some finite number of points do not lie on $P$.

Remark 1*.*

Consider $g(x)=1_{\langle x,\theta\rangle>0}\langle x,\theta\rangle^{\frac{1}{p}}$ where either $u_{i}\not\in\theta^{\perp}$ for $1\leq i\leq n$ with the assumptions of Theorem 1.6 or $u\not\in\theta^{\perp}$ for each $u\in\mathcal{P}((u_{i})_{i=1}^{m})$ in the assumptions of Theorem 1.7. Then, taking $p\to\infty$, Theorem 1.6 and Theorem 1.7 recover the results for Lebesgue measure up to a dimensional constant of $2^{n}$. The reason for this extra factor of $2^{n}$ comes from the fact that nonconstant $p-$concave, $\frac{1}{p}-$homogeneous densities are supported on at most a half-space, which therefore restricts us to only being able to get inequalities on ’half’ of our domain.

**Acknowledgements. **I am very grateful to Galyna Livshyts and Kateryna Tatarko for helpful discussions on this topic and comments on this manuscript. I would also like to thank the anonymous referee for comments that improved the exposition of this paper.

2. Extension of the Loomis-Whitney Inequality

We begin with a lemma providing us with a lower bound for the measure of a face of a parallelapiped. With homogeneity, this will give us a lower bound for the measure of a parallelapiped, which will be a key ingredient in the proof of Theorem 1.6.

Lemma 2.1**.**

Let $g,\mu,(u_{i})_{i=1}^{n}$ be as in the statement of Theorem 1.6, let

[TABLE]

where $\alpha_{1},..,\alpha_{n}$ are positive constants, and suppose that $u_{i}\in\text{supp}(g)$. Then,

[TABLE]

where $\mu_{n-1}(F_{i})$ denotes the integral of $g$ over the $(n-1)-$dimensional set $F_{i}$.

Proof.

For simplicity of notations, we deal with the case $i=1$. We begin by writing $\mu_{n-1}(F_{1})$ as an integral of $g$ over $F_{1}$, subdividing the domain of integration, and using homogeneity:

[TABLE]

If we take $\sigma^{\prime}$ such that ${\sigma}^{\prime}(j)u_{j}\in\text{supp}(g)$ for each $j$ (which can be done by the hypothesis of Theorem 1.6), then

[TABLE]

By $p-$concavity and the fact that $g(\sigma^{\prime}(j)u_{j})=\tilde{g}(u_{j})$,

[TABLE]

Inserting the bound

[TABLE]

from the arithmetic mean-geometric mean inequality under the integral gives

[TABLE]

Again by the arithmetic mean-geometric mean inequality,

[TABLE]

and thus

[TABLE]

By (2.1), our proof is complete.

∎

For the proof of our theorem, we will recall the definition of a zonotope. A zonotope is simply a Minkowski sum of line segments

[TABLE]

By linearity (1.3), if $Z=\sum_{i=1}^{m}\alpha_{i}[-u_{i},u_{i}]$ for unit vectors $u_{i}$ and $\alpha_{i}$ positive constants, then

[TABLE]

for a convex body $K$. Since our measure $\mu$ is homogeneous,

[TABLE]

by (1.4). Therefore,

[TABLE]

We now prove our theorem:

Proof of Theorem 1.6. .

Let $Z$ be the zonotope $\sum_{i=1}^{n}\alpha_{i}[-u_{i},u_{i}]$ with $\alpha_{i}=\frac{1}{P_{\mu,K}(u_{i})}$ for $1\leq i\leq n$. By Lemma 1.5, (2.2), and our choice of $\alpha_{i}$,

[TABLE]

and so

[TABLE]

Without loss of generality, we assume that $u_{i}\in\text{supp}(g)$ and $g(-u_{i})=0$ for each $i$. Let $F_{i}$ denote the face of $Z$ orthogonal to and touching $\alpha_{i}u_{i}$, and subdivide $Z$ into pyramids with bases of $F_{i}$, apex at the origin, and height of $\alpha_{i}$. By homogeneity,

[TABLE]

Applying Lemma 2.1, we have

[TABLE]

Combining this bound with (2.3) and recalling that $\alpha_{i}=\frac{1}{P_{\mu,K}(u_{i})}$, our desired inequality is proven.

∎

3. Extension of Ball’s Inequality

As in the previous section, we will require an estimate from below for the measure of a zonotope. However, mimicking the approach of Ball [Bal91], rather than estimating the measures of the faces directly, we shall first project them. A main difference from Ball’s proof stems from the lack of translation invariance of our measure, but we will circumvent this obstacle by an appropriate inequality (3.2) coming from concavity.

Lemma 3.1**.**

Let $g,\mu,(u_{i})_{i=1}^{m},(c_{i})_{i=1}^{m}$ be as in the statement of Theorem 1.7. Let $Z=\sum_{i=1}^{m}\alpha_{i}[-u_{i},u_{i}]$ be a zonotope. Then

[TABLE]

Proof.

Following Ball [Bal91], we induct on the dimension $n$. First consider the case $n=1$. We can then assume $u_{1}=...=u_{m}$ and without loss of generality $g(u_{1})=\tilde{g}(u_{1})>0$ and $g(-u_{1})=0$. Then

[TABLE]

Since $n=1$, (1.2) implies $\sum_{i=1}^{m}c_{i}=1$, and therefore by the arithmetic mean-geometric mean inequality

[TABLE]

This concludes the proof for $n=1$.

Let us assume we now have our result for dimension $n-1$, and consider the case of dimension $n$. Firstly, observe that homogeneity implies

[TABLE]

Therefore,

[TABLE]

Since $\sum_{i=1}^{m}\frac{c_{i}}{n}=1$, we use the arithmetic mean-geometric mean inequality once again to get

[TABLE]

Let $P_{i}Z$ denote the projection of $Z$ onto the hyperplane $u_{i}^{\perp}$. We wish to show

[TABLE]

where $\mu_{n-1}$ denotes integration of the density $g$ over the $(n-1)-$dimensional set $P_{i}Z$. This will compensate for the lack of translation invariance of our measure.

By assumption, one of $u_{i}$ and $-u_{i}$ lies in $\text{supp}(g)$. Without loss of generality, $u_{i}\in\text{supp}(g)$. For $w\in\mathbb{R}^{n}$ and $t>0$, concavity and homogeneity give us

[TABLE]

To be precise, concavity gives this to us when $w\in\text{supp}(g)$, but when $w\not\in\text{supp}(g)$ this is trivial. This inequality is equivalent to the statement that

[TABLE]

for any $w\in\mathbb{R}^{n}$ and $t_{1}\geq t_{2}$.

For each $w\in P_{i}Z$, let $t(w)\geq 0$ be taken so that $w+t(w)u_{i}\in\partial Z$. We now write

[TABLE]

where our integral of the density is taken over the region $(Z+[0,u_{i}])\setminus Z$. By (3.3) and continuity,

[TABLE]

This proves (3.2).

Denoting the projection of $u_{j}$ onto $u_{i}^{\perp}$ by $P_{i}(u_{j})$, we have that $P_{i}Z$ is the zonotope

[TABLE]

where $\gamma_{ji}=|u_{j}-\langle u_{i},u_{j}\rangle u_{i}|.$ A simple computation shows $\gamma_{ji}^{2}=1-\langle u_{i},u_{j}\rangle^{2}$.

We also have

[TABLE]

and this is the identity operator on $u_{i}^{\perp}$. By (3.1), (3.2), and our inductive hypothesis,

[TABLE]

From the inequality $\frac{1}{\gamma_{ji}}\geq 1$ and the relation

[TABLE]

an appropriate grouping of elements in our product completes the proof. ∎

As before, the proof of Theorem 1.7 now follows:

Proof of Theorem 1.7. .

Let $Z$ be the zonotope $\sum_{i=1}^{m}\alpha_{i}[-u_{i},u_{i}]$ where $\alpha_{i}=\frac{c_{i}}{P_{\mu,K}(u_{i})}$ for $1\leq i\leq m$. By the same argument as in the proof of Theorem 1.6, where we must use (1.2),

[TABLE]

By Lemma 3.1, we reach

[TABLE]

as desired. ∎