A Blaschke-Santal\'o inequality for unconditional log-concave measures

TL;DR

This paper discusses a variant of the Blaschke-Santaló inequality for unconditional log-concave measures, confirming its validity in this specific setting and surveying related literature.

Contribution

It proves that the Blaschke-Santaló inequality holds for all unconditional log-concave measures, extending classical geometric results to a broader measure-theoretic context.

Findings

The inequality holds for unconditional log-concave measures.

Survey of existing literature on the measure-theoretic Blaschke-Santaló inequality.

Confirmation of the inequality's validity in this measure class.

Abstract

The Blaschke-Santal\'o inequality states that the volume product $|K| \cdot |K^{o}|$ of a symmetric convex body $K \subset \mathbb{R}^n$ is maximized by the standard Euclidean unit-ball. Cordero-Erausquin asked whether the inequality remains true for all even log-concave measures. We briefly survey the literature around this question and provide details for the known fact that the inequality holds true for all unconditional log-concave measures.