The average distance property of Banach spaces
type: Item
instance of: scholarly article
title: The average distance property of Banach spaces
zbMATH Open document ID: 0881.46015
author: Pei-Kee Lin
published in: Archiv der Mathematik
publication date: 1997-07-06T00:00:00Z
review text: Let \(X\) be a real Banach space and \(S\) the unit sphere in \(X\). Then, \(X\) has the average distance property provided that there exists a unique \(\alpha>0\) satisfying the condition: \[ \forall x_1,\dots, x_n\in S\quad \exists x\in S\quad {1\over n} \sum^n_{i=1}|x_i- x|= \alpha. \] It follows from a theorem of O. Gross that every finite-dimensional Banach space has the average distance property. The author disproved the following conjecture of R. Wolf: Conjecture. If \(X\) is reflexive, then \(X\) has the average distance property. Namely, he proved that if \(1\leq p<2\), then \(L_p(0,1)\) and \(\ell_p\) do not have the average distance property (Theorem 4). Further, for a normal topological space \(K\), let \(C(K)\) be the space of all bounded continuous real functions on \(K\), with sup norm. In view of Theorem 5, the space \(C(K)\) has the average distance property if and only if \(K\) has an isolated point.
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 46B07
zbMATH DE Number: 1028221
zbMATH Keywords: average distance property
reviewed by: Maria Moszyńska
MaRDI profile type: Publication
full work available at URL: s000130050082
OpenAlex ID: W2002859514
DOI: 10.1007/S000130050082
description: scientific article; zbMATH DE number 1028221