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

  • Jairo Bochi
Article
  • 33 Downloads

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.

Keywords

Convex bodies Ellipsoids Symmetric difference metric Approximation 

Mathematics Subject Classification

52A10 90C26 51K99 

Notes

Acknowledgements

I thank Paula Porto for drawing most of the figures. I thank Włodek Kuperberg for posing the problem that motivated this paper, for telling me that ellipses with displaced centers could be discarded, for pointing to reference [20], and for suggesting Question 6.3. I am grateful to the referees for corrections, references, and criticism that allowed me to improve the paper significantly. I particularly thank one of the referees for suggesting to go beyond Question 1.1 and to consider the full family of ellipses interpolating between John and Loewner ellipses, and also for posing questions that led to the results presented in Sect. 5.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile

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