John Ellipsoids of Revolution

TL;DR

This paper introduces the concept of ellipsoids of revolution to interpolate between the problems of finding the largest Euclidean ball and the largest volume ellipsoid within a convex body, providing new optimality conditions.

Contribution

It generalizes known optimality conditions to ellipsoids of revolution and explores their properties in convex optimization problems.

Findings

Extends first-order optimality conditions to ellipsoids of revolution.

Shows extremal ellipsoids of revolution behave like classical extremal ellipsoids in certain directions.

Provides a unified framework for Euclidean balls and ellipsoids in convex bodies.

Abstract

Finding a largest Euclidean ball in a given convex body $K \subset \mathbb{R}^d$ and finding a largest volume ellipsoid in $K$ are two problems of fundamentally different nature. The first is a purely Euclidean problem, where we consider scaled copies of the origin-centered closed unit ball, whereas in the second problem, we search among all affine copies of the unit ball. In this paper, we interpolate between these two classical problems by considering ellipsoids of revolution. More generally, we study pairs of convex bodies $K$ and $L$, and seek a largest-volume affine image of $K$ contained within $L$, subject to certain restrictions on the allowed affine transformations. We derive first-order necessary conditions for optimality, generalizing known conditions from the unrestricted affine setting. Using these conditions, we show that an extremal ellipsoid of revolution exhibits properties analogous to those of either the largest-volume ellipsoid or the largest Euclidean ball, depending on whether the ellipsoid is considered along its axis of revolution or along the orthogonal complement of that axis.