Entropy-variance type and generalized Blaschke-Santaló inequalities from Courtade & Wang (2025)

I will present the article Generalized Blaschke-Santaló type inequalities without symmetry restrictions (2025) by Thomas A. Courtade and Edric Wang.

 

They prove a multi-function generalization of the Blaschke-Santaló inequality, thereby improving on the previous work of Nakamura and Tsuji (2024). They show that centered Gaussian functions saturate this general family of functional inequalities. The proof relies on a certain entropic duality. As applications, they establish:

- a Talagrand-type inequality for the Wasserstein barycenter problem, initially conjectured by Kolesnikov and Werner (Adv. Math. 2022).

- a Blaschke-Santaló-type geometric inequality for many convex bodies.

 

This is a short and fairly self-contained piece of work, which I hope to present in depth.