A generalized Legendre duality relation and Gaussian saturation

TL;DR

This paper generalizes the Legendre duality relation and proves Gaussian saturation properties related to a Blaschke--Santaló type inequality, with applications in convex geometry and optimal transportation.

Contribution

It extends the duality relation to multiple functions and establishes Gaussian saturation for a broad class of inequalities, linking Legendre duality with inverse Brascamp--Lieb inequalities.

Findings

Proved Gaussian saturation for inverse Brascamp--Lieb inequality with even, log-concave functions.

Confirmed a key case of the Kolesnikov-Werner conjecture on Blaschke--Santaló inequality.

Provided an affirmative answer to a Talagrand-type inequality conjecture involving Wasserstein barycenters.

Abstract

Motivated by the barycenter problem in optimal transportation theory, Kolesnikov--Werner recently extended the notion of the Legendre duality relation for two functions to the case for multiple functions. We further generalize the duality relation and then establish the centered Gaussian saturation property for a Blaschke--Santal\'{o} type inequality associated with it. Our approach to the understanding such a generalized Legendre duality relation is based on our earlier observation that directly links Legendre duality with the inverse Brascamp--Lieb inequality. More precisely, for a large family of degenerate Brascamp--Lieb data, we prove that the centered Gaussian saturation property for the inverse Brascamp--Lieb inequality holds true when inputs are restricted to even and log-concave functions. As an application to convex geometry, we establish the most important case of a conjecture of Kolesnikov and Werner about the Blaschke--Santal\'{o} inequality for multiple even functions as well as multiple symmetric convex bodies. Furthermore, in the direction of information theory and optimal transportation theory, this provides an affirmative answer to another conjecture of Kolesnikov--Werner about a Talagrand type inequality for multiple even probability measures that involves the Wasserstein barycenter.