Abstract and Applied Analysis
Volume 2006 (2006), Article ID 43591, 10 pages
doi:10.1155/AAA/2006/43591
Proximinality in geodesic spaces
A. Kaewcharoen1
and W.A. Kirk2
1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Department of Mathematics, University of Iowa, Iowa City 52242-1419, IA, USA
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.