Stability of the reverse Blaschke-Santalo inequality for unconditional convex bodies

TL;DR

This paper establishes a stability version of the reverse Blaschke-Santalo inequality for unconditional convex bodies, showing that bodies close to unconditional ones also satisfy the inequality, extending known results in convex geometry.

Contribution

It provides a stability result for the reverse Blaschke-Santalo inequality in the class of unconditional convex bodies, including bodies near unconditional ones.

Findings

Stability version of the reverse Blaschke-Santalo inequality proved.

Bodies close to unconditional convex bodies satisfy the inequality.

Characterization of equality cases in the stability context.

Abstract

Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two and in several special cases. In the class of unconditional convex bodies, Saint Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, which is sufficiently close to an unconditional body, satisfies the the reverse Blaschke-Santalo inequality.