The Fermat–Weber center of a planar body Q is a point in the plane from which the average distance to the points in Q is minimal. We first 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 larger than Delta(Q)/6, where Delta(Q) is the diameter of Q. From the other direction, we prove that the same average distance is at most 0.3490 Delta(Q). The new bound brings us closer to the conjectured value of Delta(Q)/3. We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.

(Joint work with Adrian Dumitrescu and Minghui Jiang)