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

Abstract

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