Ellipsoids in pseudoconvex domains

TL;DR

This work investigates maximizing the volume of Hermitian ellipsoids centered at the origin inside a bounded pseudoconvex domain $\Omega\subset\mathbb C^n$. It develops a geometric-variational framework in the ellipsoid space $\mathbb E_0$, proving existence of a maximizer and a precise boundary-measure characterization: a maximal ellipsoid $E_h$ satisfies $\int_{\partial\Omega\cap\partial E_h} h(Tz,z)\,d\mu(z)=\operatorname{tr}T$ for a suitable boundary measure $\mu$. A key geodesic convexity principle shows that ellipsoids inscribed in $\Omega$ form a convex set along geodesics in $\mathbb E_0$, yielding uniqueness when $\partial\Omega$ contains no holomorphic disc (in particular, for strongly pseudoconvex domains). The paper also presents non-uniqueness phenomena: even in the translated setting, the boundary-measure condition is not sufficient, and explicit examples show multiple maximal ellipsoids or translates can occur, highlighting subtle geometric constraints in complex John-type problems.

Abstract

We consider the problem of maximizing the volume of hermitian ellipsoids inscribed in a given pseudoconvex domain in complex Euclidean space. We prove existence and uniqueness, and give a characterization of the maximizer.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• proof : Proof of Theorem 1.3
• Proposition 4.1
• Proposition 4.2