Nearest and farthest points in spaces of curvature bounded below

Abstract

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 a 0 ∈ A such that d(x, a 0 ) = inf{d(x, a) : a ∈ A} (resp. d(x, a 0 ) = 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.

Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces

“…Recent results in the context of spaces with curvature bounded below globally were obtained in [10] where the authors proved a variant of Stečkin's Lemma and extended the results given in [16]. We include below some results obtained in [10] that constitute key tools in proving our results.…”

“…We include below some results obtained in [10] that constitute key tools in proving our results. Following [10], for κ ∈ (−∞, 0), define the real function F κ on R 3 + by…”

“…This estimation yielded a variant of Stečkin's Lemma for spaces of curvature bounded below globally. Proposition 3.9 (Espínola, Li, López [10]). Let E be a geodesic space of curvature bounded below globally by κ and let x ∈ E, r > 0, y ∈ B(x, r/2) \ {x} and 0 ≤ σ ≤ 2d(x, y).…”

“…We include below some of the results proved in [10] with the remark that the main results are much stronger and involve porosity concepts which we do not discuss in this work. Theorem 3.10 (Espínola, Li, López [10]).…”

“…A similar result was proved for the farthest point problem. Very recent results in the context of spaces with curvature bounded below globally were obtained in [10] where the authors proved a variant of Stečkin's Lemma that allowed them to give some porosity theorems which are stronger results than the ones in [16].…”