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

Abstract

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 \(\frac{1}{6} \cdot \Delta(Q)\), where Δ(Q) is the diameter of Q. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most \(\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490 \cdot \Delta(Q)\). The new bound substantially improves the previous bound of \(\frac{2}{3 \sqrt3} \cdot \Delta(Q) \approx 0.3849 \cdot \Delta(Q)\) due to Abu-Affash and Katz, and brings us closer to the conjectured value of \(\frac{1}{3} \cdot \Delta(Q)\). We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.