Heat flow regularisation, Brascamp—Lieb inequality, and convex geometry

This talk is based on the joint work with Hiroshi Tsuji (Osaka, Japan). A close link between Brascamp—Lieb inequality and convex geometry was discovered by Kieth Ball; volume ratio estimate, reverse isoperimetric problem, Buseman-Petty problem etc. In this talk we will report further link to the volume product of a convex body (Blaschke—Santalo inequality and Mahler’s conjecture). Our principal observation is to understand the volume product as a regularising effect of the Ornstein—Uhlenbeck heat flow (hypercontractivity) which is one of  archetype example of Brascamp—Lieb inequality. We then exhibit a wealth of this new link.