A Blaschke-Santaló inequality for unconditional log-concave measures

TL;DR

This work analyzes the Blaschke–Santaló inequality for even log-concave measures by proving a reduction to the unconditional setting: using Steiner symmetrizations, any symmetric convex body can be transformed into an unconditional body without decreasing the volume product with respect to an unconditional log-concave measure. It then leverages the known unconditional case to show $P_{\mu}(K) \le P_{\mu}(B)$ whenever $\mu$ is unconditional and $K$ is symmetric, thereby establishing the inequality in the unconditional regime and providing a pathway to broader cases. The results connect to generalized (B) conjectures, log-Brunn–Minkowski theory, and functional analogues, clarifying when the Euclidean ball remains extremal. The approach yields clean, elementary proofs and clarifies the status of the inequality for rotationally invariant measures and related conjectures in convex geometry.

Abstract

The Blaschke-Santaló 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.

Theorems & Definitions (7)

• corollary 1
• proof
• theorem 1
• lemma 1
• proposition 1
• proof
• proof : Proof of Lemma \ref{['lem:tou.c']}