Hypercontractivity beyond Nelson's time and its applications to Blaschke--Santal\'{o} inequality and inverse Santal\'{o} inequality

TL;DR

This paper introduces new hypercontractivity inequalities related to Ornstein--Uhlenbeck flows, establishing their implications for classical convex geometry inequalities like Blaschke--Santaló and Mahler's conjecture, and providing quantitative bounds for convex bodies.

Contribution

It develops two novel hypercontractivity inequalities that connect diffusion analysis with convex geometric inequalities, extending previous results and offering new bounds.

Findings

New hypercontractivity inequalities imply classical convex geometry inequalities.

Improved reverse hypercontractivity under symmetry conditions.

Quantitative bounds for volume products of convex bodies with curved boundaries.

Abstract

We explore an interplay between an analysis of diffusion flows such as Ornstein--Uhlenbeck flow and Fokker--Planck flow and inequalities from convex geometry regarding the volume product. More precisely, we introduce new types of hypercontractivity for the Ornstein--Uhlenbeck flow and clarify how these imply the Blaschke--Santal\'{o} inequality and the inverse Santal\'{o} inequality, also known as Mahler's conjecture. Motivated the link, we establish two types of new hypercontractivity in this paper. The first one is an improvement of Borell's reverse hypercontractivity inequality in terms of Nelson's time relation under the restriction that the inputs have an appropriate symmetry. We then prove that it implies the Blaschke--Santal\'{o} inequality. At the same time, it also provides an example of the inverse Brascamp--Lieb inequality due to Barthe--Wolff beyond their non-degenerate condition. The second one is Nelson's forward hypercontractivity inequality with exponents below 1 for the inputs which are log-convex and semi-log-concave. This yields new lower bounds of the volume product for convex bodies whose boundaries are well curved. This consequence provides a quantitative result of works by Stancu and Reisner--Sch\"{u}tt--Werner where they observed that a convex body with well curved boundary is not a local minimum of the volume product.