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.

Keywords:

• Inner product spaces
• optimal solutions
• barycenter
• level sets
• best approximation
• 1991 Mathematics Subject Classification. 41A65
• 1991 Mathematics Subject Classification. 46C15
• 1991 Mathematics Subject Classification. 52A07
• 1991 Mathematics Subject Classification. 90B85