Abstract
LetK be a convex body inR n, and letx* ∈ intK be the center of the ellipsoid of the maximal volume inscribed in the body. An arbitrary hyperplane throughx* cutsK into two convex bodiesK + andK −. We show thatw(K ±)/w(K)≤0.844..., wherew(·) is the volume of the inscribed ellipsoid.
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L. Danzer, D. Laugwitz, H. Lenz, Über das Löwnersche Ellipsoid und sein Analogen unter den einem Eikorper einbeschriebener Ellipsoiden,Arch. Math. 8 (1957).
S. P. Tarasov, L. G. Khachiyan, I. I. Erlich, The method of inscribed ellipsoids,Soviet Math. Dokl. 37(1) (1988).
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Khachiyan, L.G. An inequality for the volume of inscribed ellipsoids. Discrete Comput Geom 5, 219–222 (1990). https://doi.org/10.1007/BF02187786
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DOI: https://doi.org/10.1007/BF02187786