Extremal inscribed and circumscribed complex ellipsoids

TL;DR

This paper investigates the properties of extremal inscribed and circumscribed complex ellipsoids within convex sets in complex space, establishing uniqueness and translation relations, and extending classical geometric characterizations to complex settings.

Contribution

It proves the uniqueness of minimal circumscribed complex ellipsoids and characterizes extremal inscribed ellipsoids, extending classical convex geometry results to complex spaces.

Findings

Uniqueness of minimal circumscribed complex ellipsoid.

If two inscribed maximal volume ellipsoids exist, they are translates.

Extension of Brunn's characterization to complex ellipsoids.

Abstract

We prove that if a convex set in Cn contains two inscribed complex ellipsoid of maximal volume then one is a translate of the other. On the other hand, the circumscribed complex elipsoid of minimal volume is unique. As application we prove the complex analoge of Brunn's characterization of ellipsods.