A Note on Geodesically Bounded R-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 Gamma, and if a nonexpansive mapping T : X -> Gamma satisfies inf{d (x, T(x)) x is an element of X} = 0, then T has a fixed point. The latter result fails if T is only continuous.