Chebyshev centers and some geometric properties of Banach spaces

In this paper, we study two properties (P1) and (P2) introduced in the literature for studying the existence and stability of relative Chebyshev centers. We reformulate these properties in a way that leads naturally to our results. We mainly relate these properties with some geometric properties of Banach spaces. In particular, reflexive spaces having the Kadec-Klee property are characterized in terms of property (P1) and uniform convexity is characterized in terms of property (P2). Some continuity results for the center map are also presented.