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From simplex slicing to sharp reverse Hölder inequalities

James Melbourne, Michael Roysdon, Colin Tang, Tomasz Tkocz

TL;DR

The paper extends Webb's sharp simplex slicing bound to a probabilistic setting by establishing sharp reverse Hölder-type inequalities for centred log-concave random variables using negative moments. It introduces an $L_1$ proxy in place of an $L_2$ constraint and reduces the optimization to the one-parameter family of two-sided exponential densities, then proves a phase transition in the extremising distribution at a critical value $p_0$ with explicit constants. The main results yield a tight $L_p$-$L_1$ inequality: for $-1<p\le 1$, $\|X\|_p \ge \Gamma(p+1)^{1/p}\|X\|_1$, and for $p\ge 1$, $\|X\|_p \le C_p\|X\|_1$ with $C_p$ determined by $p_0$, attaining equality at double-exponential or one-sided exponential distributions. These findings connect convex geometric questions about simplex sections to sharp probabilistic moment inequalities and suggest explicit extremisers and constants with potential implications for geometric tomographic problems.

Abstract

Simplex slicing (Webb, 1996) is a sharp upper bound on the volume of central hyperplane sections of the regular simplex. We extend this to sharp bounds in the probabilistic framework of negative moments, and beyond, of centred log-concave random variables, establishing a curious phase transition of the extremising distribution for new sharp reverse Hölder-type inequalities.

From simplex slicing to sharp reverse Hölder inequalities

TL;DR

The paper extends Webb's sharp simplex slicing bound to a probabilistic setting by establishing sharp reverse Hölder-type inequalities for centred log-concave random variables using negative moments. It introduces an proxy in place of an constraint and reduces the optimization to the one-parameter family of two-sided exponential densities, then proves a phase transition in the extremising distribution at a critical value with explicit constants. The main results yield a tight - inequality: for , , and for , with determined by , attaining equality at double-exponential or one-sided exponential distributions. These findings connect convex geometric questions about simplex sections to sharp probabilistic moment inequalities and suggest explicit extremisers and constants with potential implications for geometric tomographic problems.

Abstract

Simplex slicing (Webb, 1996) is a sharp upper bound on the volume of central hyperplane sections of the regular simplex. We extend this to sharp bounds in the probabilistic framework of negative moments, and beyond, of centred log-concave random variables, establishing a curious phase transition of the extremising distribution for new sharp reverse Hölder-type inequalities.
Paper Structure (17 sections, 12 theorems, 89 equations, 1 figure)

This paper contains 17 sections, 12 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results.
  4. Related works.
  5. Proofs
  6. Overall strategy and the main idea: $L_1$ proxy
  7. Crossing arguments.
  8. A reduction to two-sided exponentials.
  9. Auxiliary lemmata
  10. Proof of \ref{['eq:E_t-p<1']}
  11. Proof of \ref{['eq:E_t-p>1']}
  12. The whole range $p \geq 1$ reduces to $p = p_0$.
  13. Proof of \ref{['eq:E_t-p>1']} at transition point $p=p_0$
  14. Final remarks
  15. Variance constraint.
  16. ...and 2 more sections

Key Result

Theorem 1

For every $-1 < p \leq 1$ and every mean $0$ log-concave random variable $X$, we have which is sharp with equality attained by a standard double-exponential random variable $X$ (with density $\frac{1}{2}e^{-|x|}$ on $\mathbb{R}$).

Figures (1)

  • Figure 1: The intersection pattern of $\log f$ and $\log g$.

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Remark 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 13 more