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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Affine diameters of convex bodies
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by Imre Bárány, Daniel Hug and Rolf Schneider PDF
Proc. Amer. Math. Soc. 144 (2016), 797-812 Request permission

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.
  • MR Author ID: 30885
  • Email: barany.imre@renyi.mta.hu
  • Daniel Hug
  • Affiliation: Department of Mathematics, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany
  • MR Author ID: 363423
  • Email: daniel.hug@kit.edu
  • Rolf Schneider
  • Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, D-79104 Freiburg i. Br., Germany
  • MR Author ID: 199426
  • ORCID: 0000-0003-0039-3417
  • Email: rolf.schneider@math.uni-freiburg.de
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 797-812
  • MSC (2010): Primary 52A20, 52A40; Secondary 46B20
  • DOI: https://doi.org/10.1090/proc12746
  • MathSciNet review: 3430855