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- W2963017862 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 $$Ecap K$$ : 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." @default.
- W2963017862 created "2019-07-30" @default.
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- W2963017862 date "2018-06-20" @default.
- W2963017862 modified "2023-10-18" @default.
- W2963017862 title "On the Approximation of Convex Bodies by Ellipses with Respect to the Symmetric Difference Metric" @default.
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- W2963017862 doi "https://doi.org/10.1007/s00454-018-0015-z" @default.
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