New Bounds on the Average Distance from the Fermat-Weber Center of a Planar Convex Body

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 of Q is larger than 16 · ∆(Q), where ∆(Q) is the diameter of Q. This proves a conjecture of Carmi, HarPeled and Katz. From the other direction, we prove that the same average distance is at most 2(4− √ 3) 13 ·∆(Q) < 0.3490 ·∆(Q). The new bound substantially improves the previous bound of 2 3 √ 3 ·∆(Q) ≈ 0.3849 ·∆(Q) due to Abu-Affash and Katz, and brings us closer to the conjectured value of 1 3 ·∆(Q). We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.