Localization in multifacility location theory

We present two localization theorems for a multifacility location problem involving linear costs and a single norm in a two-dimensional space. It is shown that optimal locations for all the new facilities can be found in the metric hull of existing facilities, whatever the norm is. In the polyhedral norm case we study if only finite set of intersections points which belong to the metric hull needs to be considered; the result is obtained in particular for the ℓ¹-norm and a counter example is given for a general polyhedral norm.