Minimal volume product of convex bodies with certain discrete symmetries and its applications

TL;DR

This paper establishes the minimal volume product for convex bodies with certain symmetries, advancing understanding of Mahler's conjecture and its implications in symplectic geometry.

Contribution

It provides the sharp lower bounds for volume products of symmetric convex bodies, extending partial results related to Mahler's conjecture and Viterbo's conjecture.

Findings

Sharp lower bounds for volume products of symmetric convex bodies.

Partial progress on Mahler's conjecture for specific symmetries.

Insights into Viterbo's conjecture from a convex geometric perspective.

Abstract

We give the sharp lower bound of the volume product of $n$-dimensional convex bodies which are invariant under a discrete subgroup $SO(K)=\{ g \in SO(n); g(K)=K \}$, where $K$ is an $n$-cube or $n$-simplex. This provides new partial results of Mahler's conjecture and its non-symmetric version. In addition, we give partial answers for Viterbo's isoperimetric type conjecture in symplectic geometry from the view point of Mahler's conjecture.