A Note on Geodesically Bounded -Trees

It is proved that a complete geodesically bounded R-tree is the closed convex hull of the set of its extreme points. It is also noted that if X is a closed convex geodesically bounded subset of a complete R-tree Y, and if a nonexpansive mapping T:X→Y satisfies inf {d(x,T(x)):x∈X}=0, then T has a fixed point. The latter result fails if T is only continuous.