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This paper addresses the general continuous single facility location problemsin finite dimension spaces under possibly different $\ell_p$ norms in thedemand points. We analyze the difficulty of this family of problems and revisitconvergence properties of some well-known algorithms. The ultimate goal is toprovide a common approach to solve the family of continuous $\ell_p$ orderedmedian location problems in dimension $d$ (including of course the $\ell_p$minisum or Fermat-Weber location problem for any $p\ge 1$). We prove that thisapproach has a polynomial worse case complexity for monotone lambda weights andcan be also applied to constrained and even non-convex problems.

Revisiting several problems and algorithms in continuous location with $\ell_p$ norms