Minimal volume product of three dimensional convex bodies with various discrete symmetries

TL;DR

This paper determines the minimal volume product for 3D convex bodies with certain symmetries, advancing understanding of Mahler's conjecture in non-symmetric cases.

Contribution

It provides sharp lower bounds and characterizations for volume products of symmetric convex bodies in three dimensions, contributing new partial results to Mahler's conjecture.

Findings

Sharp lower bounds for volume products under discrete symmetries

Characterizations of convex bodies with minimal volume product

Progress towards non-symmetric Mahler's conjecture in 3D

Abstract

We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under a discrete subgroup of $O(3)$ in several cases. We also characterize the convex bodies with the minimal volume product in each case. In particular, this provides a new partial result of the non-symmetric version of Mahler's conjecture in the three dimensional case.