The General One Center Location Problem

The general one center location problem deals with the location of a point in a real normed space X in order to minimize an objective function G which depends on the distances to a finite number of centers and on initial costs. The function G is defined by G(x) = γ(c₁ + w₁‖x − a₁‖, …, c_n + w_n‖x − a_n‖), where a₁, …, a_n are n given points in X, w₁, …, w_n are positive numbers, c₁, …, c_n are nonnegative initial costs and γ is a monotone norm on ℝⁿ. A geometrical description of the set of optimal solutions to the problem min_x∈XG(x) is provided. The peculiar role of the minisum problem, where γ is the l₁-norm, is emphasized and the minimax problem, where γ is the l_x-norm, is used to illustrate the general geometrical description.