Sharp comparison of moments and the log-concave moment problem

TL;DR

This paper establishes sharp moment comparison results for various classes of random variables, including optimal constants in Khintchine inequalities for vectors in $ ext{ell}_q^n$ and characterizes extremisers of moments for symmetric log-concave functions.

Contribution

It provides the first sharp constants for Khintchine inequalities for $q>2$ and characterizes extremisers of moments for symmetric log-concave functions, refining existing theorems.

Findings

Optimal constants in Khintchine inequalities for $q>2$

Extremal properties of weighted sums of symmetric uniform distributions

Characterization of extremisers of moments for symmetric log-concave functions

Abstract

This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors uniformly distributed on the unit ball of the space $\ell_q^n$ for $q\in(2,\infty)$, complementing past works that treated $q\in(0,2]\cup\{\infty\}$. As a byproduct of this result, we prove an extremal property for weighted sums of symmetric uniform distributions among all symmetric unimodal distributions. In the second part we provide a one-to-one correspondence between vectors of moments of symmetric log-concave functions and two simple classes of piecewise log-affine functions. These functions are shown to be the unique extremisers of the $p$-th moment functional, under the constraint of a finite number of other moments being fixed, which is a refinement of the description of extremisers provided by the generalised localisation theorem of Fradelizi and Gu\'edon [Adv. Math. 204 (2006) no. 2, 509-529].