Efficiency in Euclidean constrained location problems

In this note we present geometrical characterizations for the set of efficient, weakly efficient and properly efficient solutions to the multiobjective Euclidean Location problem with convex locational constraints, extending the known results for the unconstrained problem. It is shown that the set of the (weakly) efficient points coincides with the closest-point projection of the convex hull of the demand points onto the feasible set S. It is also shown that the set of properly efficient solutions is the union of two sets: the set of feasible demand points and the closest-point projection of the relative interior of the convex hull of the demand points onto S