On a Santaló point for Nakamura-Tsuji's Laplace transform inequality

Abstract

Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santaló 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.

35 pages, added new facts concerning convergence of integrals of log-concave functions (Facts 2.4 and 2.5) and the limit as p to 0 (Fact 2.10). Expanded references to other recent works