Minimal volume product of three dimensional convex bodies invariant under certain groups of order four

TL;DR

This paper establishes the minimal volume product for certain three-dimensional convex bodies invariant under specific order-four symmetry groups, advancing understanding of Mahler's conjecture in non-symmetric cases.

Contribution

It provides the sharp lower bounds and characterizations of convex bodies with minimal volume product under two types of order-four symmetry groups in three dimensions.

Findings

Identified sharp lower bounds for volume products.

Characterized convex bodies achieving minimal volume product.

Contributed partial results towards non-symmetric Mahler's conjecture.

Abstract

We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under two kinds of discrete subgroups of $O(3)$ of order four. We also characterize the convex bodies with the minimal volume product in each case. This provides new partial results of the non-symmetric version of Mahler's conjecture in the three dimensional case.