The non-symmetric Mahler conjecture in dimension three

TL;DR

This paper proves the non-symmetric Mahler conjecture in three dimensions, establishing a sharp lower bound for the volume product of convex bodies with respect to the Santaló point.

Contribution

It provides a proof of the non-symmetric Mahler conjecture in dimension three, confirming the conjectured lower bound for the volume product.

Findings

Established the sharp lower bound rac{64}{9}or the volume product in or convex bodies in or rom the conjecture.

Confirmed the conjecture specifically in three-dimensional space.

Advances understanding of volume products in convex geometry.

Abstract

We prove the non-symmetric Mahler conjecture in dimension three. More precisely, we prove the sharp lower bound \[ \mathcal P(K) \geq \frac{64}{9} \] for every convex body $K \subset \mathbb R^3$, where $\mathcal P(K)$ denotes the non-symmetric volume product with respect to the Santal\'o point.