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Abstract and Applied Analysis
Volume 2006, Article ID 43591, 10 pages
http://dx.doi.org/10.1155/AAA/2006/43591

Proximinality in geodesic spaces

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA

Received 4 May 2006; Revised 1 August 2006; Accepted 10 August 2006

Copyright © 2006 A. Kaewcharoen and W. A. Kirk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let X be a complete CAT(0) space with the geodesic extension property and Alexandrov curvature bounded below. It is shown that if C is a closed subset of X, then the set of points of X which have a unique nearest point in C is Gδ and of the second Baire category in X. If, in addition, C is bounded, then the set of points of X which have a unique farthest point in C is dense in X. A proximity result for set-valued mappings is also included.