New bounds on the average distance from the Fermat–Weber center of a planar convex body

The Fermat–Weber center of a planar body QQ is a point in the plane from which the average distance to the points in QQ is minimal. We first show that for any convex body QQ in the plane, the average distance from the Fermat–Weber center of QQ to the points in QQ is larger than 16⋅Δ(Q), where Δ(Q)Δ(Q) is the diameter of QQ. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most 2(4−3)13⋅Δ(Q)<0.3490⋅Δ(Q). The new bound substantially improves the previous bound of 233⋅Δ(Q)≈0.3849⋅Δ(Q) due to Abu-Affash and Katz, and brings us closer to the conjectured value of 13⋅Δ(Q). We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.