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 mappingT∗T^*of a metric treeMMwith convex values has a selectionT:M→MT: M\rightarrow Mfor whichd(T(x),T(y))≤dH(T∗(x),T∗(y))d(T(x),T(y))\leq d_H(T^*(x),T^*(y))for eachx,y∈Mx,y \in M. Here bydHd_Hwe mean the Hausdroff distance. Many applications of this result are given.