Local minimality of the volume-product at the simplex

TL;DR

This paper proves that the simplex is a strict local minimum for the volume product among convex bodies, extending stability results and methods to the non-symmetric setting in convex geometry.

Contribution

It establishes the local minimality of the volume product at the simplex and extends stability methods to non-symmetric convex bodies.

Findings

The simplex is a strict local minimum for the volume product.

Linear local stability near the simplex is demonstrated.

Methods are extended to non-symmetric convex bodies.

Abstract

It is proved that the simplex is a strict local minimum for the volume product, P(K)=min(vol(K) vol(K^z)), K^z is the polar body of K with respect to z, the minimum is taken over z in the interior of K, in the Banach-Mazur space of n-dimensional (classes of ) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest, concerning stability of square order of volumes of polars of non-symmetric convex bodies.