On the mean width of log-concave functions

TL;DR

This paper introduces a new definition for the mean width of log-concave functions, establishes its equivalence with previous definitions, and derives several geometric inequalities and estimates related to it.

Contribution

It proposes a natural new definition for the mean width of log-concave functions and proves key inequalities and estimates, including a functional Urysohn inequality and volume ratio bounds.

Findings

New definition of mean width coincides with previous one

Derived a functional Urysohn inequality

Proved finite volume ratio and low-M* estimates

Abstract

In this work we present a new, natural, definition for the mean width of log-concave functions. We show that the new definition coincide with a previous one by B. Klartag and V. Milman, and deduce some properties of the mean width, including an Urysohn type inequality. Finally, we prove a functional version of the finite volume ratio estimate and the low-M* estimate.