On Pareto optima, the Fermat-Weber problem, and polyhedral gauges

This paper deals with multiobjective programming in which the objective functions are nonsymmetric distances (derived from different gauges) to the points of a fixed finite subset of ℝⁿ. It emphasizes the case in which the gauges are polyhedral. In this framework the following result is known: if the gauges are polyhedral, then each Pareto optimum is the solution to a Fermat-Weber problem with strictly positive coefficients. The paper gives a new proof of this result, and it shows that it is useful in finding the whole set of efficient points of a location problem with polyhedral gauges. Also, the paper characterizes polyhedral gauges in terms of a property of their subdifferential.