On convex metrics

Abstract

Given a finit e number of points PI, ., PI." in th e plane, consider the proble m of findin g a point x that minimizes the sum of Euclidean distances 'Ld(p; , x) . More general versions of thi s proble m arise in s patial economics, concerning opti mal locations for a central office , plant, or ware ho use (compare [3]). Mos t of these will be based on me trics d more general than th e Euclidean metric. Among them, th e class of metri cs that are convex fun ctions in each variable co mmand partic ular interest: in thi s case, local minima are automati cally global minima, facilitating minimization decisively. We s hall show in thi s paper that convex metrics are invariant under translation, and therefore arise from a norm. For th e concepts of topologies, metrics , and norms in linear spaces see, for instance, [1)2 and [2]. 1. Metrics and norms. Le t L be a linear s pace over the fi eld R of real numbers. A fun ction f: L -7 R is convex if