On the stochastic proof of the Blaschke-Santaló inequality

2025/12/09 by Lehec, Joseph
Mathematics · #Mathematical Inequalities and Applications #Point processes and geometric inequalities #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2512.08454

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 Prékopa-Leindler inequality. The symmetrized Talagrand inequality can be seen as a dual form of the functional Santaló 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 Santaló inequality directly, rather than in its dual form.

Citations

• Gaussian Correlation via Inverse Brascamp-Lieb
• A generalized Legendre duality relation and Gaussian saturation
• Stochastic proof of the sharp symmetrized Talagrand inequality
• The functional volume product under heat flow
• A sharp symmetrized form of Talagrand's transport-entropy inequality for the Gaussian measure
• Positive Gaussian kernels also have Gaussian minimizers
• Representation formula for the entropy and functional inequalities
• Some functional forms of Blaschke-Santaló inequality
• Some deviation inequalities

Related

• On the functional Blaschke-Santalo inequality
• Stochastic proof of the sharp symmetrized Talagrand inequality
• Symmetrized Talagrand Inequalities on Euclidean Spaces
• Stability of the functional forms of the Blaschke-Santalo inequality
• Some extensions of the Pr 'ekopa-Leindler inequality using Borell's\n stochastic approach