Generalized Blaschke--Santal\'o-type inequalities, without symmetry restrictions

Abstract

Nakamura and Tsuji (2024) recently investigated a many-function generalization of the functional Blaschke--Santal\'o inequality, which they refer to as a generalized Legendre duality relation. They showed that, among the class of all even test functions, centered Gaussian functions saturate this general family of functional inequalities. Leveraging a certain entropic duality, we give a short alternate proof of Nakamura and Tsuji's result, and, in the process, eliminate all symmetry assumptions. As an application, we establish a Talagrand-type inequality for the Wasserstein barycenter problem (without symmetry restrictions) originally conjectured by Kolesnikov and Werner (\textit{Adv.~Math.}, 2022). An analogous geometric Blaschke--Santal\'o-type inequality is established for many convex bodies, again without symmetry assumptions.