On the Set of Optimal Points to the Weber Problem

It is known that the planar Weber location problem with l_p distances has all its solutions in the convex hull of the demand points. for l₁ and l_∞ distances, additional conditions are known which reduce the set of possible optimal points to the intersection of that convex hull, the efficient set, and the points defined by a certain grid. In this paper, we determine the smallest set which includes at least one optimal point for every Weber problem based on a given set of demand points. It is shown that for 1<p<∞ a certain part of the convex hull is the smallest possible set, but for p=1 or p=∞ the known conditions do not necessarily yield the correct set. Finally, we find the smallest possible set for p=1 or p=∞.