A remark on the Mahler conjecture: local minimality of the unit cube

TL;DR

This paper proves that the unit cube is a strict local minimizer of the Mahler volume product among symmetric convex bodies, using the Banach-Mazur distance, contributing to the understanding of the Mahler conjecture.

Contribution

It establishes the local minimality of the unit cube for the Mahler volume product in the class of symmetric convex bodies.

Findings

The unit cube is a strict local minimizer for the Mahler volume product.

The proof uses the Banach-Mazur distance to analyze local minimality.

Supports the Mahler conjecture by identifying local minimizers.

Abstract

We prove that the unit cube $B^n_{\infty}$ is a strict local minimizer for the Mahler volume product $vol_n(K)vol_n(K^*)$ in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.