Nearest and farthest points in spaces of curvature bounded below

Let A be a nonempty closed subset (resp. nonempty bounded closed subset) of a metric space (X, d) and x ∈ X \ A. The nearest point problem (resp. the farthest point problem) w.r.t. x considered here is to find a point a0 ∈ A such that d(x, a0) = inf{d(x, a) : a ∈ A} (resp. d(x, a0) = sup{d(x, a) : a ∈ A}). We study the well posedness of nearest point problems and farthest point problems in geodesic spaces. We show that if X is a space of curvature bounded below then the complement of the set of all points x ∈ X for which the nearest point problem (resp. the farthest point problem) w.r.t. x is well posed is σ -porous in X \ A. Our results extend and/or improve some recent results about proximinality in geodesic spaces as well as the corresponding ones previously obtained in uniformly convex Banach spaces. In particular, the result regarding the nearest point problem answers affirmatively an open problem proposed by Kaewcharoen and Kirk [A. Kaewcharoen, W.A. Kirk, Proximinality in geodesic spaces, Abstr. Appl. Anal. 2006 (2006) 1–10]. c © 2010 Elsevier Inc. All rights reserved.