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

Given a centrally symmetric convex body K⊂ R^d and a positive number λ, we consider, among all ellipsoids E⊂ R^d of volume λ, those that best approximate K with respect to the symmetric difference metric, or equivalently that maximize the volume of E∩ 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 λ 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∩ E is a strictly quasiconcave function of the ellipse E, with respect to the natural affine structure on the set of ellipses of area λ. 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.