Affine diameters of convex bodies

Bárány, Imre and Hug, Daniel and Schneider, Rolf (2016) Affine diameters of convex bodies. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 144 (2). pp. 797-812. ISSN 0002-9939

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Abstract

We prove sharp inequalities for the average number of affine diameters through the points of a convex body K in Rn. These inequalities hold if K is a polytope or of dimension two. An example shows that the proof given in the latter case does not extend to higher dimensions. The example also demonstrates that for n ≥ 3 there exist norms and convex bodies K ⊂ Rn such that the metric projection on K with respect to the metric defined by the given norm is well defined but not a Lipschitz map, which is in striking contrast to the planar or the Euclidean case. © 2015 American Mathematical Society.

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