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

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 inequality corresponding to the conjecture is sometimes called the reverse Blaschke-Santaló inequality. The conjecture is known to be true in R 2 and for 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 reverse Blaschke-Santaló inequality.

Equipartitions and Mahler volumes of symmetric convex bodies

“…The same method as in the proof of Theorem 3, i.e. applying Lemma 5, known equality cases and the results from [19,32] can be used to present shorter proofs of the stability theorems given in [8,20].…”