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 . The new bound substantially improves the previous bound of due to Abu-Affash and Katz, and brings us closer to the conjectured value of . We also confirm the upper bound conjecture for centrally symmetric planar convex bodies. © 2009 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Dumitrescu, A., & Tóth, C. D. (2009). New bounds on the average distance from the Fermat-Weber center of a planar convex body. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 5878 LNCS, pp. 132–141). https://doi.org/10.1007/978-3-642-10631-6_15
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