Optimal Locations and Inner Products

In a normed space X, we consider objective functions which depend on the distances between a variable point and the points of certain finite sets A. A point where such a function attains its minimum on X is generically called an optimal location. In this paper we obtain characterizations of inner product spaces with properties connecting optimal locations and the convex hull of A or barycenters of points of A with well chosen weights. We thus generalize several classical results about characterization of inner product spaces