https://yale.zoom.us/j/95303636613 [4]
This talk is based on joint works with Hiroshi Tsuji (Saitama Japan).
The Blaschke—Santal\'{o} inequality describes a correlation between a convex body and its dual object (polar body). Motivated by the recent studies in convex geometry, optimal transportation theory, as well as information theory, a problem of extending the inequality to multiple convex bodies was proposed by Kolesnikov--Werner. Their formulation of the problem naturally involves some generalization of the (functional) Legendre duality. In this talk, we are going to establish a genuine Gaussian saturation principle for the generalized Blaschke--Santal\'{o}-type inequality, and in particular give an affirmative answer to the conjecture of Kolesnikov--Werner.

Our novel observation is a simple but crucial link between the above problem and the inverse form of the Brascamp--Lieb (multilinear) inequality (IBL inequality). The study of the IBL inequality was initiated by Chen--Dafnis--Paouris, and then later Barthe--Wolff developed its theory in more general framework, but under a certain non-degeneracy condition.
Our second main result is about the Gaussian saturation principle for the IBL inequality beyond the framework of Barthe—Wolff.
The above result on the generalized Blaschke—Santal\’{o}-type inequality is a consequence of this second result.
There are further fruitful consequences from our study of the IBL inequality, which we will present as long as time permits.