An overview of complex ellipsoids

TL;DR

This paper reviews the properties of complex ellipsoids within convex geometry, highlighting recent progress such as the proof of the Complex Banach Conjecture, and discusses their mathematical significance.

Contribution

It provides a comprehensive overview of complex ellipsoids and includes the recent proof of the Complex Banach Conjecture, a significant advancement in the field.

Findings

Proof of the Complex Banach Conjecture

Characterizations of complex ellipsoids in convex geometry

Comparison with real ellipsoids

Abstract

An ellipsoid is the image of a ball under an affine transformation. If this affine transformation is over the complex numbers, we refer to it as a complex ellipsoid. Characterizations of real ellipsoids have received much attention over the years however, characterizations of complex ellipsoids have been studied very little. This paper is a review of what is known about complex ellipsoids from the point of view of convex geometry. In particular, the proof of the Complex Banach Conjecture.