Isotropic constants and Mahler volumes

TL;DR

This paper explores the relationships between convex body volumes, isotropic constants, and Mahler volumes, establishing new implications, bounds, and counterexamples in convex geometry.

Contribution

It demonstrates that a sharp Bourgain slicing conjecture implies the Mahler conjecture for non-symmetric bodies and shows how translating polars affects isotropic constants.

Findings

A sharp Bourgain slicing conjecture implies the Mahler conjecture.

Translating the polar of a convex body can bound its isotropic constant.

Counter-example to Kuperberg's volume distribution conjecture.

Abstract

This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.