Projective transformations of convex bodies and volume product

Abstract

In this paper we study certain variational aspects of the volume product functional restricted to the space of small projective deformations of a fixed convex body. In doing so, we provide a short proof of a theorem by Klartag: a strong version of the slicing conjecture implies the Mahler conjecture. We also exhibit an interesting family of critical convex bodies in dimension $2$ containing saddle points for this functional.