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Averaging distances in finite dimensional normed spaces and John's ellipsoid


Author: Aicke Hinrichs
Journal: Proc. Amer. Math. Soc. 130 (2002), 579-584
MSC (2000): Primary 52A21, 46B04
DOI: https://doi.org/10.1090/S0002-9939-01-06160-3
Published electronically: May 25, 2001
MathSciNet review: 1862140
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Abstract: 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

\begin{displaymath}\frac{1}{n}\sum_{k=1}^n\Vert x_k-x\Vert=r.\end{displaymath}

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 $X$.


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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: nah@rz.uni-jena.de

DOI: https://doi.org/10.1090/S0002-9939-01-06160-3
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
Article copyright: © Copyright 2001 American Mathematical Society

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