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Extremal convex bodies for affine measures of symmetry. (English) Zbl 1472.52012

M. Meyer et al. introduced in [Adv. Math. 228, No. 5, 2920–2942 (2011; Zbl 1241.52003)] affine measures of symmetry for a convex body \(K\) based on a pair \((p_1(K),p_2(K))\) of affine covariant points in \(K\), e.g., the centroid \(g(K)\), the Santaló point \(s(K)\) or the John point \(j(K)\) and Löwner point \(l(K)\), which are the centers of the John and Löwner ellipsoid, respectively. In the paper under review the author gives bounds for this measure in the cases \((g(K),j(K))\), \((g(K),l(K))\). For the pair \((j(K),l(K))\) see also [O. Mordhorst, Isr. J. Math. 219, No. 2, 529–548 (2017; Zbl 1376.52015)].

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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References:

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