New bounds on the average distance from the Fermat-Weber center of a planar convex body
英文摘要：
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 ${1/6} \cdot Δ(Q)$, 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 $\frac{2(4-\sqrt3)}{13} \cdot Δ(Q) < 0.3490 \cdot Δ(Q)$. The new bound substantially improves the previous bound of $\frac{2}{3 \sqrt3} \cdot Δ(Q) \approx 0.3849 \cdot Δ(Q)$ due to Abu-Affash and Katz, and brings us closer to the conjectured value of ${1/3} \cdot Δ(Q)$. We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.
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