A Selection Theorem in Metric Trees

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Mathematics (CMC)

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In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping T of a metric tree M with convex values has a selection T : M → M for which d(T(x), T(y)) ≤ dH(T(x), T(y)) for each x, y ∈ M. Here by dH we mean the Hausdroff distance. Many applications of this result are given.

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© 2006 American Mathematical Society