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

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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 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.