Best Approximation in ℝ-Trees

Abstract

An ℝ-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. We give a constructive proof of the following “best approximation” theorem in such spaces. Suppose X is a closed convex and geodesically bounded subset of an ℝ-tree H, and suppose T:X → 2 ^H is a multivalued upper semicontinuous mapping whose values are nonempty closed convex subsets of X. Then there exists a point x ₀ ∊ X such that We also give a topological version of the above theorem in a more abstract setting, and we prove a KKM theorem for geodesically bounded ℝ-trees.