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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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Averaging distances in finite dimensional normed spaces and John’s ellipsoid
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Proc. Amer. Math. Soc. 130 (2002), 579-584 Request permission


A Banach space $X$ has the average distance property (ADP) if there exists a unique real number $r=r(X)$ such that for each positive integer $n$ and all $x_1,\ldots ,x_n$ in the unit sphere of $X$ there is some $x$ in the unit sphere of $X$ such that \[ \frac {1}{n}\sum _{k=1}^n\| x_k-x\|=r.\] A theorem of Gross implies that every finite dimensional normed space has the average distance property. We show that, if $X$ has dimension $d$, then $r(X) \le 2-1/d$. This is optimal and answers a question of Wolf (Arch. Math., 1994). The proof is based on properties of the John ellipsoid of maximal volume contained in the unit ball of $X$.
  • Marco Baronti, Emanuele Casini, and Pier Luigi Papini, On the average distance property and the size of the unit sphere, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), no. suppl., 427–446. Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996). MR 1645732
  • Joan Cleary, Sidney A. Morris, and David Yost, Numerical geometry—numbers for shapes, Amer. Math. Monthly 93 (1986), no. 4, 260–275. MR 835294, DOI 10.2307/2323675
  • O. Gross, The rendezvous value of metric space, Advances in Game Theory, Princeton Univ. Press, Princeton, N.J., 1964, pp. 49–53. MR 0162643
  • Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
  • Nicole Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 993774
  • Reinhard Wolf, On the average distance property of spheres in Banach spaces, Arch. Math. (Basel) 62 (1994), no. 4, 338–344. MR 1264706, DOI 10.1007/BF01201787
  • Reinhard Wolf, On the average distance property in finite-dimensional real Banach spaces, Bull. Austral. Math. Soc. 51 (1995), no. 1, 87–101. MR 1313116, DOI 10.1017/S0004972700013927
  • R. Wolf, Averaging distances in real quasihypermetric Banach spaces of finite dimension, Isr. J. Math., 110, 125–151, 1999.
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Additional Information
  • Aicke Hinrichs
  • Affiliation: Mathematisches Institut, FSU Jena, D 07743 Jena, Germany
  • Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • Email:
  • Received by editor(s): July 24, 2000
  • Published electronically: May 25, 2001
  • Additional Notes: The author was supported by DFG grant Hi 584/2-1.
  • Communicated by: N. Tomczak-Jaegermann
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 579-584
  • MSC (2000): Primary 52A21, 46B04
  • DOI:
  • MathSciNet review: 1862140