Simple proof of the lower bound on the average distance from the Fermat-Weber center of a convex body

Abstract

We show that for any convex body Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at least Δ(Q)/6, where Δ(Q) denotes the diameter of Q. Our proof is simple and straightforward, since it needs only elementary calculations. This simplifies a previously known proof that is based on Steiner symmetrizations.