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 | References | Similar Articles | Additional Information

Abstract: A Banach space has the average distance property (ADP) if there exists a unique real number such that for each positive integer and all in the unit sphere of there is some in the unit sphere of such that

A theorem of Gross implies that every finite dimensional normed space has the average distance property. We show that, if has dimension , then . 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 .

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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