A Lower Bound for the Mahler Volume of Symmetric Convex Sets

TL;DR

This paper establishes a new lower bound for the Mahler volume of symmetric convex bodies in dimensions four and higher, using a dimension-dependent constant derived from a variational procedure and localisation techniques.

Contribution

It introduces a novel computable constant and a proof method for lower bounds on Mahler volume, extending understanding in convex geometry.

Findings

Derived a lower bound for Mahler volume in ≥4 dimensions

Defined a dimension-dependent constant via a 2D variational procedure

Applied localisation techniques similar to Gromov's Waist of the Sphere theorem

Abstract

The goal of this paper is to present a lower bound for the Mahler volume of at least 4-dimensional symmetric convex bodies. We define a computable dimension dependent constant through a 2-dimensional variational (max-min) procedure and demonstrate that the Mahler volume of every (at least 4-dimensional) symmetric convex body is greater than a (simple) function of this constant. Similar to the proof of Gromov's Waist of the Sphere Theorem in [18], our result is proved via localisation-type arguments obtained from a suitable measurable partition (or partitions) of the canonical sphere.