A selection theorem in metric trees

Abstract

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 ∗ T^* of a metric tree M M with convex values has a selection T : M → M T: M\rightarrow M for which d ( T ( x ) , T ( y ) ) ≤ d H ( T ∗ ( x ) , T ∗ ( y ) ) d(T(x),T(y))\leq d_H(T^*(x),T^*(y)) for each x , y ∈ M x,y \in M . Here by d H d_H we mean the Hausdroff distance. Many applications of this result are given.