Improved bounds on the average distance to the Fermat–Weber center of a convex object

We show that for any convex object Q in the plane, the average distance between the Fermat–Weber center of Q and the points in Q is at least 4Δ(Q)/25, and at most , where Δ(Q) is the diameter of Q. We use the former bound to improve the approximation ratio of a load-balancing algorithm of Aronov et al. [B. Aronov, P. Carmi, M.J. Katz, Minimum-cost load-balancing partitions, Algorithmica, in press].