A generalized Legendre duality relation and Gaussian saturation

Abstract Motivated by the barycenter problem in optimal transportation theory, Kolesnikov–Werner recently extended the concept of the Legendre duality relation from two functions to multiple functions. We further generalize this duality relation and then establish the centered Gaussian saturation property for a Blaschke–Santaló-type inequality associated with it. Our approach to understanding this generalized Legendre duality relation is based on the 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 when inputs are restricted to even and log-concave functions. As an application to convex geometry, we establish a significant case of a conjecture by Kolesnikov–Werner concerning the Blaschke–Santaló inequality for multiple even functions and multiple symmetric convex bodies. Furthermore, in the context of information theory and optimal transportation theory, this provides an affirmative answer to another conjecture by Kolesnikov–Werner concerning a Talagrand-type inequality for multiple even probability measures involving the Wasserstein barycenter.