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 in Q is larger than 1 6 ⋅ Δ ( 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 2 ( 4 − 3 ) 13 ⋅ Δ ( Q ) < 0.3490 ⋅ Δ ( Q ) . The new bound substantially improves the previous bound of 2 3 3 ⋅ Δ ( Q ) ≈ 0.3849 ⋅ Δ ( Q ) due to Abu-Affash and Katz, and brings us closer to the conjectured value of 1 3 ⋅ Δ ( Q ) . We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.