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

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

2010-02-01

arxiv.org/abs/1002.0345v1

Mathematics

Metric Geometry

13 pages, 2 figures. An earlier version (now obsolete): A. Dumitrescu and Cs. D. T\'oth: New bounds on the average distance fr

Scientific paper

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 ${1/6} \cdot \Delta(Q)$, where $\Delta(Q)$ 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 ${1/3} \cdot \Delta(Q)$. We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.

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