# Ellipsoids are the only local maximizers of the volume product

## TL;DR

This paper proves that ellipsoids are the only local maximizers of the volume product among convex bodies, establishing their global optimality using shadow systems and Steiner symmetrization techniques.

## Contribution

The paper demonstrates that ellipsoids uniquely maximize the volume product locally and globally, extending previous results with new geometric methods.

## Key findings

- Ellipsoids are the only local maximizers of the volume product.
- Local maximizers are also global maximizers, confirming ellipsoids' optimality.
- Uses shadow systems and Steiner symmetrization in the proof.

## Abstract

Using previous results about shadow systems and Steiner symmetrization, we prove that the local maximizers of the volume product of convex bodies are actually the global maximizers, that is: ellipsoids.