A Lower Bound for the Mahler Volume of Symmetric Convex Sets

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.