On the Approximation of Convex Bodies by Ellipses with Respect to the Symmetric Difference Metric

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 $I_K(\mathord {\cdot })$. 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.