John’s Ellipsoid and the Integral Ratio of a Log-Concave Function

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.

The Löwner Function of a Log-Concave Function

“…Let ψ be a convex function. Let S z be as in (1). Then (i) L and L z are involutions, that is, L(Lψ) = ψ and L z (L z ψ) = ψ.…”

“…It was only recently that the notion of a John (ellipsoid) function of a log-concave function was established by Alonso-Gutiérrez, Merino, Jiménez, and Villa [1]. However, the notion of a Löwner ellipsoid function for log-concave functions has been missing till now.…”

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We introduce the notion of Löwner (ellipsoid) function for a log-concave function and show that it is an extension of the Löwner ellipsoid for convex bodies. We investigate its duality relation to the recently defined John (ellipsoid) function [1]. For convex bodies, John and Löwner ellipsoids are dual to each other.

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A Reverse Rogers–Shephard Inequality for Log-Concave Functions

“…We will use the following definition of the polar projection body of f which involves level sets, equivalent to the one stated in the introduction (see [AGJV,Proposition 4.1]).…”

The functional inequality for the mixed quermassintegral

“…The affine isoperimetric inequality for the log-concave function is proved by Artstein-Avidan, Klartag, Schütt and Werner [5]. The John ellipsoid for log-concave function has been establish by Gutiérrez, Merino Jiménez and Villa [2], the LYZ ellipsoid for log-concave function is established by Fang and Zhou [16]. See [1,4,9,[12][13][14]23] for more about the pertinent results.…”