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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
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.

References [Enhancements On Off] (What's this?)

  • [1] Imre Bárány and Tudor Zamfirescu, Diameters in typical convex bodies, Canad. J. Math. 42 (1990), no. 1, 50-61. MR 1043510 (91a:52002),
  • [2] Herbert Busemann, The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955. MR 0075623 (17,779a)
  • [3] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325 (41 #1976)
  • [4] P. C. Hammer, Diameters of convex bodies, Proc. Amer. Math. Soc. 5 (1954), 304-306. MR 0061398 (15,819b)
  • [5] Preston C. Hammer and Andrew Sobczyk, Planar line families. II, Proc. Amer. Math. Soc. 4 (1953), 341-349. MR 0056942 (15,149e)
  • [6] D. Hug, Measures, curvatures and currents in convex geometry, Habilitationsschrift, Albert-Ludwigs-Universität, Freiburg, 1999.
  • [7] C. A. Rogers and G. C. Shephard, Convex bodies associated with a given convex body, J. London Math. Soc. 33 (1958), 270-281. MR 0101508 (21 #318)
  • [8] Rolf Schneider, Polytopes and Brunn-Minkowski theory, Polytopes: abstract, convex and computational (Scarborough, ON, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 440, Kluwer Acad. Publ., Dordrecht, 1994, pp. 273-299. MR 1322067 (96a:52016)
  • [9] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
  • [10] V. Soltan, Affine diameters of convex-bodies--a survey, Expo. Math. 23 (2005), no. 1, 47-63. MR 2133336 (2005k:52013),
  • [11] Rolf Walter, Some analytical properties of geodesically convex sets, Abh. Math. Sem. Univ. Hamburg 45 (1976), 263-282. MR 0417984 (54 #6029)

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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.

Daniel Hug
Affiliation: Department of Mathematics, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany

Rolf Schneider
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, D-79104 Freiburg i. Br., Germany

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

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