Convex hull properties in location theory

We present new characterizations of inner product spaces which bring into play a property of a family of optimization problems related to the norm of the space. This property concerns the existence of a solution .to some optimization problems which belongs to the convex hull of some set. We thus obtain a generalization of results of V. Klee and A. Garkavi about the Chebychev centers and also of more recent results of the author about Fermat points. Intermediate propositions concerning unicity in some optimization problems, a geometric characterization of finite dimensional inner product spaces and monotone norms seem to have their own interest.