A Selection Theorem in Metric Trees
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.
© 2006 American Mathematical Society
A.Aksoy, M. A. Khamsi, “A selection theorem in metric trees” Proc. of Amer .Math. Soc. 134 (2006) No. 10, 2957-2966.