John's ellipsoid and the integral ratio of a log-concave function

TL;DR

This paper extends John's ellipsoid concept to log-concave functions, introduces the integral ratio, and establishes a reverse affine isoperimetric inequality, providing new insights into functional geometric inequalities.

Contribution

It defines the integral ratio for log-concave functions and identifies the maximizing function, extending classical volume ratio concepts to a functional setting.

Findings

Identification of the log-concave function maximizing the integral ratio

Establishment of a reverse functional affine isoperimetric inequality

Extension of volume ratio concepts to integrable log-concave functions

Abstract

We extend the notion of John's ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.