Constrained and unconstrained problems in location theory and inner products

Abstract

In a real normed space X the optimization problem associated to a finite subset and to a family of positive weights with the objective function [UM0001] has some well known properties when X is an inner product space:1. The set of optimal solutions to the unconstrained problem min[math003] is reduced to the barycenter g of the weighted points (a$sub:i$esub:, W$sub:i$esub:). 2. The nonempty level sets of G are spheres centered at g. 3. The set of optimal solutions to the constrained problem min[math004], where Ω is a nonempty closed proper subset of X, is the best approximation set for g in ω. This paper studies converse statements. It establishes characterizations of inner product spaces among the class of normed spaces with properties of one of the above type.