From simplex slicing to sharp reverse H\"older inequalities

TL;DR

This paper extends the concept of simplex slicing to establish sharp reverse H"older inequalities for centered log-concave random variables, revealing a phase transition in extremising distributions.

Contribution

It introduces a novel probabilistic framework for simplex slicing, deriving sharp bounds for negative moments of log-concave variables and identifying a phase transition in extremisers.

Findings

Established sharp bounds for negative moments of log-concave variables

Discovered a phase transition in extremising distributions

Extended simplex slicing concepts to probabilistic inequalities

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 H\"older-type inequalities.