On the approximation of convex bodies by ellipses with respect to the symmetric difference metric

TL;DR

This paper investigates the uniqueness of maximal intersection ellipsoids approximating convex bodies in the plane, proving uniqueness in 2D by showing the intersection volume function is strictly quasiconcave.

Contribution

It establishes the uniqueness of MI ellipses in two dimensions and characterizes maximal intersection positions under transversality assumptions.

Findings

Proves uniqueness of MI ellipses in 2D.

Shows the intersection volume function is strictly quasiconcave.

Provides a characterization of maximal intersection positions.

Abstract

Given a centrally symmetric convex body $K \subset \mathbb{R}^d$ and a positive number $\lambda$, we consider, among all ellipsoids $E \subset \mathbb{R}^d$ of volume $\lambda$, those that best approximate $K$ with respect to the symmetric difference metric, or equivalently that maximize the volume of $E\cap K$: these are the maximal intersection (MI) ellipsoids introduced by Artstein-Avidan and Katzin. The question of uniqueness of MI ellipsoids (under the obviously necessary assumption that $\lambda$ is between the volumes of the John and the Loewner ellipsoids of $K$) is open in general. We provide a positive answer to this question in dimension $d=2$. Therefore we obtain a continuous $1$-parameter family of ellipses interpolating between the John and the Loewner ellipses of $K$. In order to prove uniqueness, we show that the area $I_K(E)$ of the intersection $K \cap E$ is a strictly quasiconcave function of the ellipse $E$, with respect to the natural affine structure on the set of ellipses of area $\lambda$. The proof relies on smoothening $K$, putting it in general position, and obtaining uniform estimates for certain derivatives of the function $I_K(.)$. Finally, we provide a characterization of maximal intersection positions, that is, the situation where the MI ellipse of $K$ is the unit disk, under the assumption that the two boundaries are transverse.