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

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.