An arithmetic proof of John's ellipsoid theorem

TL;DR

This paper presents a clear proof of John's ellipsoid theorem using Voronoi's ideas, offering insights into the geometric properties of maximal volume ellipsoids within convex bodies.

Contribution

It introduces a transparent proof of John's theorem based on Voronoi's approach, and extends the idea to a generalized context highlighting its challenges.

Findings

Proof of John's ellipsoid theorem using Voronoi's ideas

Extension of the approach to a generalized version of John's theorem

Identification of difficulties in the 'easy part' of the generalization

Abstract

Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John's characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the 'hard part' of a generalization of John's theorem and shows the difficulties of the corresponding 'easy part'.