A generalized Legendre duality relation and Gaussian saturation

This talk is based on the joint work with Hiroshi Tsuji (Saitama, Japan). 

Motivated by the Wasserstein barycenter problem, Kolesnikov-Werner recently extended  notions of the polar duality of convex bodies and the Legendre duality for functions to those of multiple inputs. Based on these notions, they formulated the multiple-input extension of the Blaschke--Santal\'{o} inequality and the symmetric Talagrand inequality for Wasserstein barycenter. They then proved these inequalities for unconditional convex bodies and unconditional functions.  

In this talk, we confirm that these inequalities hold under the evenness assumption. 

Our proof is based on an observation that these Blaschke--Santal\'{o}-type inequalities may be regarded as the limiting case of so-called inverse Brascamp--Lieb inequality. This family of inequalities were introduced by Chen--Dafnis--Paouris, and then further investigated by Barthe--Wolff.