A Remark on Measures of Sections of $\boldsymbol{L}_{p}$-balls

Abstract

We prove that there exists an absolute constant C so that

$$\displaystyle{ \mu (K)\ \leq \ C\sqrt{p}\max _{\xi \in S^{n-1}}\mu (K \cap \xi ^{\perp })\ \vert K\vert ^{1/n} }$$

for any p > 2, any $n \in \mathbb{N},$ any convex body K that is the unit ball of an n-dimensional subspace of L _p , and any measure μ with non-negative even continuous density in $\mathbb{R}^{n}.$ Here ξ ^⊥ is the central hyperplane perpendicular to a unit vector ξ ∈ S ⁿ⁻¹, and | K | stands for volume.

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1 Introduction

The slicing problem [1, 4, 5, 29], a major open question in convex geometry, asks whether there exists a constant C so that for any $n \in \mathbb{N}$ and any origin-symmetric convex body K in $\mathbb{R}^{n},$

$$\displaystyle{ \vert K\vert ^{\frac{n-1} {n} } \leq C\max _{\xi \in S^{n-1}}\vert K \cap \xi ^{\perp }\vert, }$$

where | K | stands for volume of proper dimension, and ξ ^⊥ is the central hyperplane in $\mathbb{R}^{n}$ perpendicular to a unit vector ξ. The best-to-date result C ≤ O(n ^1∕4) is due to Klartag [15], who improved an earlier estimate of Bourgain [6]. The answer is affirmative for unconditional convex bodies (as initially observed by Bourgain; see also [3, 14, 29]), intersection bodies [10, Theorem 9.4.11], zonoids, duals of bodies with bounded volume ratio [29], the Schatten classes [23], k-intersection bodies [21, 22]; see [7] for more details.

The case of unit balls of finite dimensional subspaces of L _p is of particular interest in this note. It was shown by Ball [2] that the slicing problem has an affirmative answer for the unit balls of finite dimensional subspaces of L _p ,  1 ≤ p ≤ 2. Junge [13] extended this result to every p ∈ (1, ∞), with the constant C depending on p and going to infinity when p → ∞. Milman [27] gave a different proof for subspaces of L _p ,  2 < p < ∞, with the constant $C \leq O(\sqrt{p}).$ Another proof of this estimate can be found in [22].

A generalization of the slicing problem to arbitrary measures was considered in [18–21]. Does there exist a constant C so that for every n ∈ N, every origin-symmetric convex body K in $\mathbb{R}^{n},$ and every measure μ with non-negative even continuous density f in $\mathbb{R}^{n},$

$$\displaystyle{ \mu (K)\ \leq \ C\max _{\xi \in S^{n-1}}\mu (K \cap \xi ^{\perp })\ \vert K\vert ^{1/n}? }$$

(1)

For every k-dimensional subspace of $\mathbb{R}^{n},\ 1 \leq k \leq n$ and any Borel set A ⊂ E, 

$$\displaystyle{ \mu (A) =\int _{A}f(x)dx, }$$

where the integration is with respect to the k-dimensional Lebesgue measure on E. 

Inequality (1) was proved with an absolute constant C for intersection bodies [18] (see [16], this includes the unit balls of subspaces of L _p with 0 < p ≤ 2), unconditional bodies and duals of bodies with bounded volume ratio in [20], for k-intersection bodies in [21]. For arbitrary origin-symmetric convex bodies, (1) was proved in [19] with $C \leq O(\sqrt{n}).$ A different proof of the latter estimate was recently given in [8], where the symmetry condition was removed.

For the unit balls of subspaces of L _p ,  p > 2,  (1) was proved in [21] with C ≤ O(n ^{1∕2−1∕p}). In this note we improve the estimate to $C \leq O(\sqrt{p}),$ extending Milman’s result [27] to arbitrary measures in place of volume. In fact, we prove a more general inequality

$$\displaystyle{ \mu (K)\ \leq \ (C\sqrt{p})^{k}\max _{ H\in Gr_{n-k}}\mu (K \cap H)\ \vert K\vert ^{k/n}, }$$

(2)

where 1 ≤ k < n,  Gr _n−k is the Grassmanian of (n − k)-dimensional subspaces of $\mathbb{R}^{n},\ K$ is the unit ball of any n-dimensional subspace of L _p ,  p > 2, μ is a measure on $\mathbb{R}^{n}$ with even continuous density, and C is a constant independent of p, n, k, K, μ. 

The proof is a combination of two known results. Firstly, we use the reduction of the slicing problem for measures to computing the outer volume ratio distance from a body to the class of intersection bodies established in [20]; see Proposition 1. Note that outer volume ratio estimates have been applied to different cases of the original slicing problem by Ball [2], Junge [13], and Milman [27]. Secondly, we use an estimate for the outer volume ratio distance from the unit ball of a subspace of L _p ,  p > 2, to the class of origin-symmetric ellipsoids proved by Milman in [27]. This estimate also follows from results of Davis, Milman and Tomczak-Jaegermann [9]. We include a concentrated version of the proof in Proposition 2.

2 Slicing Inequalities

We need several definitions and facts. A closed bounded set K in $\mathbb{R}^{n}$ is called a star body if every straight line passing through the origin crosses the boundary of K at exactly two points, the origin is an interior point of K, and the Minkowski functional of K defined by

$$\displaystyle{ \|x\|_{K} =\min \{ a \geq 0:\ x \in aK\} }$$

is a continuous function on $\mathbb{R}^{n}.$

The radial function of a star body K is defined by

$$\displaystyle{ \rho _{K}(x) =\| x\|_{K}^{-1},\qquad x \in \mathbb{R}^{n},\ x\neq 0. }$$

If x ∈ S ⁿ⁻¹ then ρ _K (x) is the radius of K in the direction of x. 

We use the polar formula for volume of a star body

$$\displaystyle{ \vert K\vert = \frac{1} {n}\int _{S^{n-1}}\|\theta \|_{K}^{-n}d\theta. }$$

(3)

The class of intersection bodies was introduced by Lutwak [25]. Let K, L be origin-symmetric star bodies in $\mathbb{R}^{n}.$ We say that K is the intersection body of L if the radius of K in every direction is equal to the (n − 1)-dimensional volume of the section of L by the central hyperplane orthogonal to this direction, i.e. for every ξ ∈ S ⁿ⁻¹, 

$$\displaystyle\begin{array}{rcl} \rho _{K}(\xi )& =& \|\xi \|_{K}^{-1} = \vert L \cap \xi ^{\perp }\vert {}\\ & =& \frac{1} {n - 1}\int _{S^{n-1}\cap \xi ^{\perp }}\|\theta \|_{L}^{-n+1}d\theta = \frac{1} {n - 1}R\left (\|\cdot \|_{L}^{-n+1}\right )(\xi ), {}\\ \end{array}$$

where R: C(S ⁿ⁻¹) → C(S ⁿ⁻¹) is the spherical Radon transform

$$\displaystyle{ Rf(\xi ) =\int _{S^{n-1}\cap \xi ^{\perp }}f(x)dx,\qquad \forall f \in C(S^{n-1}). }$$

All bodies K that appear as intersection bodies of different star bodies form the class of intersection bodies of star bodies. A more general class of intersection bodies is defined as follows. If μ is a finite Borel measure on S ⁿ⁻¹, then the spherical Radon transform R μ of μ is defined as a functional on C(S ⁿ⁻¹) acting by

$$\displaystyle{ (R\mu,f) = (\mu,Rf) =\int _{S^{n-1}}Rf(x)d\mu (x),\qquad \forall f \in C(S^{n-1}). }$$

A star body K in $\mathbb{R}^{n}$ is called an intersection body if ∥ ⋅ ∥  _K ⁻¹ = R μ for some measure μ, as functionals on C(S ⁿ⁻¹), i.e.

$$\displaystyle{ \int _{S^{n-1}}\|x\|_{K}^{-1}f(x)dx =\int _{ S^{n-1}}Rf(x)d\mu (x),\qquad \forall f \in C(S^{n-1}). }$$

Intersection bodies played a crucial role in the solution of the Busemann-Petty problem and its generalizations; see [17, Chap. 5].

A generalization of the concept of an intersection body was introduced by Zhang [30] in connection with the lower dimensional Busemann-Petty problem. For 1 ≤ k ≤ n − 1, the (n − k)-dimensional spherical Radon transform R _n−k : C(S ⁿ⁻¹) → C(Gr _n−k ) is a linear operator defined by

$$\displaystyle{ R_{n-k}g(H) =\int _{S^{n-1}\cap H}g(x)\ dx,\quad \forall H \in Gr_{n-k} }$$

for every function g ∈ C(S ⁿ⁻¹). 

We say that an origin symmetric star body K in $\mathbb{R}^{n}$ is a generalized k -intersection body, and write $K \in \mathcal{B}\mathcal{P}_{k}^{n},$ if there exists a finite Borel non-negative measure μ on Gr _n−k so that for every g ∈ C(S ⁿ⁻¹)

$$\displaystyle{ \int _{S^{n-1}}\|x\|_{K}^{-k}g(x)\ dx =\int _{ Gr_{n-k}}R_{n-k}g(H)\ d\mu (H). }$$

(4)

When k = 1 we get the class of intersection bodies. It was proved by Goodey and Weil [11] for k = 1 and by Grinberg and Zhang [12, Lemma  6.1] for arbitrary k (see also [28] for a different proof) that the class $\mathcal{B}\mathcal{P}_{k}^{n}$ is the closure in the radial metric of k-radial sums of origin-symmetric ellipsoids. In particular, the classes $\mathcal{B}\mathcal{P}_{k}^{n}$ contain all origin-symmetric ellipsoids in $\mathbb{R}^{n}$ and are invariant with respect to linear transformations. Recall that the k-radial sum K + _k L of star bodies K and L is defined by

$$\displaystyle{ \rho _{K+_{k}L}^{k} =\rho _{ K}^{k} +\rho _{ L}^{k}. }$$

For a convex body K in $\mathbb{R}^{n}$ and 1 ≤ k < n, denote by

$$\displaystyle{ \text{o.v.r.}(K,\mathcal{B}\mathcal{P}_{k}^{n}) =\inf \left \{\left ( \frac{\vert C\vert } {\vert K\vert }\right )^{1/n}:\ K \subset C,\ C \in \mathcal{B}\mathcal{P}_{ k}^{n}\right \} }$$

the outer volume ratio distance from a body K to the class $\mathcal{B}\mathcal{P}_{k}^{n}.$

Let B ₂ ⁿ be the unit Euclidean ball in $\mathbb{R}^{n},$ let | ⋅ | ₂ be the Euclidean norm in $\mathbb{R}^{n},$ and let σ be the uniform probability measure on the sphere S ⁿ⁻¹ in $\mathbb{R}^{n}.$ For every $x \in \mathbb{R}^{n}$, let x ₁ be the first coordinate of x. We use the fact that for every p > −1

$$\displaystyle{ \int _{S^{n-1}}\vert x_{1}\vert ^{p}d\sigma (x) = \frac{\Gamma (\frac{p+1} {2} )\Gamma (\frac{n} {2} )} {\sqrt{\pi }\Gamma (\frac{n+p} {2} )}; }$$

(5)

see for example [17, Lemma 3.12], where one has to divide by $\vert S^{n-1}\vert = 2\pi ^{(n-1)/2}/\Gamma (\frac{n} {2} ),$ because the measure σ on the sphere is normalized.

In [20], the slicing problem for arbitrary measures was reduced to estimating the outer volume ratio distance from a convex body to the classes $\mathcal{B}\mathcal{P}_{k}^{n}$, as follows.

Proposition 1

For any $n \in \mathbb{N},\ 1 \leq k <n,$ any origin-symmetric star body K in $\mathbb{R}^{n},$ and any measure μ with even continuous density on K,

$$\displaystyle{ \mu (K) \leq \left (\text{ o.v.r.}(K,\mathcal{B}\mathcal{P}_{k}^{n})\right )^{k}\ \frac{n} {n - k}c_{n,k}\max _{H\in Gr_{n-k}}\mu (K \cap H)\ \vert K\vert ^{k/n}, }$$

where c _n,k = |B ₂ ⁿ | ^(n−k)∕n ∕|B ₂ ^n−k |∈ (e ^−k∕2 ,1).

It appears that for the unit balls of subspaces of L _p ,  p > 2 the outer volume ration distance to the classes of intersection bodies does not depend on the dimension. As mentioned in the introduction, the following estimate was proved in [27] and also follows from results of [9]. We present a short version of the proof.

Proposition 2

Let $p> 2,\ n \in \mathbb{N},\ 1 \leq k <n,$ and let K be the unit ball of an n-dimensional subspace of L _p . Then

$$\displaystyle{ \text{ o.v.r.}(K,\mathcal{B}\mathcal{P}_{k}^{n}) \leq C\sqrt{p}, }$$

where C is an absolute constant.

Proof

Since the classes $\mathcal{B}\mathcal{P}_{k}^{n}$ are invariant under linear transformations, we can assume that K is in the Lewis position. By a result of Lewis in the form of [26, Theorem 8.2], this means that there exists a measure ν on the sphere so that for every $x \in \mathbb{R}^{n}$

$$\displaystyle{ \|x\|_{K}^{p} =\int _{ S^{n-1}}\vert (x,u)\vert ^{p}d\nu (u), }$$

and

$$\displaystyle{ \vert x\vert _{2}^{2} =\int _{ S^{n-1}}\vert (x,u)\vert ^{2}d\nu (u). }$$

Also, by the same result of Lewis [24], K ⊂ n ^{1∕2−1∕p} B ₂ ⁿ. 

Let us estimate the volume of K from below. By the Fubini theorem, formula (5) and Stirling’s formula, we get

$$\displaystyle\begin{array}{rcl} \int _{S^{n-1}}\|x\|_{K}^{p}d\sigma (x)& =& \int _{ S^{n-1}}\int _{S^{n-1}}\vert (x,u)\vert ^{p}d\sigma (x)d\nu (u) {}\\ & =& \int _{S^{n-1}}\vert x_{1}\vert ^{p}d\sigma (x)\int _{ S^{n-1}}d\nu (u) \leq \left ( \frac{Cp} {n + p}\right )^{p/2}\int _{ S^{n-1}}d\nu (u). {}\\ \end{array}$$

Now

$$\displaystyle\begin{array}{rcl} \frac{Cp} {n + p}\left (\int _{S^{n-1}}d\nu (u)\right )^{2/p}& \geq & \left (\int _{ S^{n-1}}\|x\|_{K}^{p}d\sigma (x)\right )^{2/p} {}\\ & \geq & \left (\int _{S^{n-1}}\|x\|_{K}^{-n}d\sigma (x)\right )^{-2/n} = \left ( \frac{\vert K\vert } {\vert B_{2}^{n}\vert }\right )^{-2/n} \sim \frac{1} {n}\vert K\vert ^{-2/n}, {}\\ \end{array}$$

because | B ₂ ⁿ | ^1∕n ∼ n ^−1∕2. On the other hand,

$$\displaystyle\begin{array}{rcl} 1& =& \int _{S^{n-1}}\vert x\vert _{2}^{2}d\sigma (x) =\int _{ S^{n-1}}\int _{S^{n-1}}(x,u)^{2}d\nu (u)d\sigma (x) {}\\ & =& \int _{S^{n-1}}\int _{S^{n-1}}\vert x_{1}\vert ^{2}d\sigma (x)d\nu (u) = \frac{1} {n}\int _{S^{n-1}}d\nu (u), {}\\ \end{array}$$

so

$$\displaystyle{ \frac{Cp} {n + p}n^{2/p} \geq \frac{1} {n}\vert K\vert ^{-2/n}, }$$

and

$$\displaystyle{ \vert K\vert ^{1/n} \geq cn^{-1/p}\sqrt{\frac{n + p} {np}} \geq \frac{cn^{1/2-1/p}} {\sqrt{p}} \vert B_{2}^{n}\vert ^{1/n}. }$$

Finally, since K ⊂ n ^{1∕2−1∕p} B ₂ ⁿ, and $B_{2}^{n} \in \mathcal{B}\mathcal{P}_{k}^{n}$ for every k, we have

$$\displaystyle{ \text{ o.v.r.}(K,\mathcal{B}\mathcal{P}_{k}^{n}) \leq \left (\frac{\vert n^{1/2-1/p}B_{ 2}^{n}\vert } {\vert K\vert } \right )^{1/n} \leq C\sqrt{p}, }$$

where C is an absolute constant.

We now formulate the main result of this note.

Corollary 1

There exists a constant C so that for any $p> 2,\ n \in \mathbb{N},\ 1 \leq k <n,$ any convex body K that is the unit ball of an n-dimensional subspace of L _p , and any measure μ with non-negative even continuous density in $\mathbb{R}^{n},$

$$\displaystyle{ \mu (K)\ \leq \ (C\sqrt{p})^{k}\max _{ H\in Gr_{n-k}}\mu (K \cap H)\ \vert K\vert ^{k/n}. }$$

Proof

Combine Proposition 1 with Proposition 2. Note that $\frac{n} {n-k} \in (1,e^{k}),$ and c _n, k  ∈ (e ^−k∕2, 1), so these constants can be incorporated in the constant C. □