On a Santal\'o point for Nakamura-Tsuji's Laplace transform inequality

TL;DR

This paper extends a Laplace transform inequality related to the Blaschke-Santaló inequality from even to non-even functions using a novel centering approach and explores its geometric implications and applications.

Contribution

It introduces a new method for extending Laplace transform inequalities to non-even functions via a centering procedure based on translations.

Findings

Extension of the inequality to non-even functions.

New insights into the geometry of the Laplace transform.

Application to reverse hypercontractivity.

Abstract

Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santal\'o inequality in its functional form. Inspired by their method, based on the Fokker-Planck semi-group, we extend the inequality to non-even functions. We consider a well-chosen centering procedure by studying the infimum over translations in a double Laplace transform. This requires a new look on the existing methods and leads to several observations of independent interest on the geometry of the Laplace transform. Application to reverse hypercontractivity is also given.