Asymptotic behavior of the Weber location problem on the plane

The asymptotic behavior of the Weber location problem is investigated. The authors consider problems were n demand points are randomly generated in a unit disk by a uniform distribution and all weights are equal to one. The main result of the paper is that the probability that the optimal solution be on a demand point is approximately 1/n. Additional results for a large n: the optimal solution converges almost surely to the center of the disk; the difference between the optimal value of the objective function and the minimal value of the objective function on a demand point converges to 1/2.