On the stochastic proof of the Blaschke-Santal\'o inequality

TL;DR

This paper simplifies the stochastic proof of the functional Santaló inequality, originally proved via a dual form of the symmetrized Talagrand inequality, by modifying Borell's original argument.

Contribution

It provides a streamlined, accessible proof of the functional Santaló inequality using a simple modification of Borell's stochastic approach.

Findings

Simplified stochastic proof of the functional Santaló inequality.

Direct derivation of the inequality from Borell's original method.

Clarification of the relationship between Talagrand and Santaló inequalities.

Abstract

In 2024, Courtade, Fathi and Mikulincer gave a proof of the symmetrized Talagrand inequality based on stochastic calculus, in the spirit of Borell's proof of the Pr\'ekopa-Leindler inequality. The symmetrized Talagrand inequality can be seen as a dual form of the functional Santal\'o inequality. The modest purpose of this note is to give a simplified version of the Courtade, Fathi and Mikulincer argument. Namely we first recall briefly Borell's original argument, and we then explain a simple twist in his proof that allows to recover the functional Santal\'o inequality directly, rather than in its dual form.