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Affine diameters of convex bodies


Authors: Imre Bárány, Daniel Hug and Rolf Schneider
Journal: Proc. Amer. Math. Soc. 144 (2016), 797-812
MSC (2010): Primary 52A20, 52A40; Secondary 46B20
DOI: https://doi.org/10.1090/proc12746
Published electronically: May 28, 2015
MathSciNet review: 3430855
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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 $ {\mathbb{R}}^n$. 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\ge 3$ there exist norms and convex bodies $ K\subset \mathbb{R}^n$ 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.


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Additional Information

Imre Bárány
Affiliation: Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary – and – Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom.
Email: barany.imre@renyi.mta.hu

Daniel Hug
Affiliation: Department of Mathematics, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany
Email: daniel.hug@kit.edu

Rolf Schneider
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, D-79104 Freiburg i. Br., Germany
Email: rolf.schneider@math.uni-freiburg.de

DOI: https://doi.org/10.1090/proc12746
Received by editor(s): March 25, 2014
Received by editor(s) in revised form: December 12, 2014, and January 4, 2015
Published electronically: May 28, 2015
Additional Notes: This research was partially supported by ERC Advanced Research Grant no 267165 (DISCONV). The first author was supported by Hungarian National Foundation Grant K 83767. The second author was partially supported by the German Research Foundation (DFG) under the grant HU 1874/4-2.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society