A note on “On Pareto optima, the Fermat-Weber problem, and polyhedral gauges”

We show by counterexample that one of the results in the paper On Pareto optima, the Fermat-Weber problem, and polyhedral gauges, by R. Durier, Mathematical Programming 47 (1990), does not hold. The stated characterization of properly efficient points for a scalar location problem is only true in dimension 1 and 2. For higher dimensions, the property stated is necessary, but not sufficient.