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

Abstract

Let A and X be nonempty, bounded and closed subsets of a geodesic metric space (E, d). The minimization (resp. maximization) problem denoted by min(A, X) (resp. max(A, X)) consists in finding (a0,x0)∈A×X(a0,x0)∈A×X such that d(a0,x0)=inf{d(a,x):a∈A,x∈X}d(a0,x0)=inf{d(a,x):a∈A,x∈X} (resp. d(a0,x0)=sup{d(a,x):a∈A,x∈X}d(a0,x0)=sup{d(a,x):a∈A,x∈X}). We give generic results on the well-posedness of these problems in different geodesic spaces and under different conditions considering the set A fixed. Besides, we analyze the situations when one set or both sets are compact and prove some specific results for CAT(0) spaces. We also prove a variant of the Drop Theorem in Busemann convex geodesic spaces and apply it to obtain an optimization result for convex functions.Dirección General de Enseñanza SuperiorJunta de AntalucíaThe Sectoral Operational Programme Human Resources Developmen

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“…A geodesic space X is said to be reflexive if for every decreasing chain {C α } ⊆ X with α ∈ I such that C α is nonempty, bounded, closed and convex for all α ∈ I we have that α∈I C α = ∅. It was announced in [8] that a reflexive and Busemann convex space is complete. The following property of reflexive and Busemann convex spaces plays an important role in our coming discussions.…”

“…For example consider X = R 2 with radial metric defined with d (x 1 , y 1 ), (x 2 , y 2 ) =    ρ (x 1 , y 1 ), (x 2 , y 2 ) ; if (0, 0), (x 1 , y 1 ), (x 2 , y 2 ) are colinear, ρ (x 1 , y 1 ), (0, 0) + ρ (x 2 , y 2 ), (0, 0) ; otherwise, where ρ denotes the usual Euclidean metric on R 2 . Then (X, d) is a complete Rtree and so is a reflexive and Busemann convex space (see [8] for more details). Note that the radial metric does not induced with any norm.…”

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