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 \({(a_0,x_0) \in A \times X}\) such that \({d(a_0,x_0) = \inf\left\{d(a,x) : a \in A, x \in X\right\}}\) (resp. \({d(a_0,x_0) = \sup\left\{d(a,x) : a \in A, x \in X\right\}}\)). 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.
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Espínola, R., Nicolae, A. Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces. Monatsh Math 165, 173–197 (2012). https://doi.org/10.1007/s00605-010-0266-0
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DOI: https://doi.org/10.1007/s00605-010-0266-0