The ultimate planar convex hull algorithm
SIAM Journal on Computing
The algebraic degree of geometric optimization problems
Discrete & Computational Geometry
Fast approximations for sums of distances, clustering and the Fermat--Weber problem
Computational Geometry: Theory and Applications
Efficient algorithms for detecting regular point configurations
ICTCS'05 Proceedings of the 9th Italian conference on Theoretical Computer Science
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In this paper, we study the properties of the Fermat-Weber point for a set of fixed points, whose arrangement coincides with the vertices of a regular polygonal chain. A k-chain of a regular n-gon is the segment of the boundary of the regular n-gon formed by a set of k (≤ n) consecutive vertices of the regular n-gon. We show that for every odd positive integer k, there exists an integer N(k), such that the Fermat-Weber point of a set of k fixed points lying on the vertices a k-chain of a n-gon coincides with a vertex of the chain whenever n ≥ N(k). We also show that $\lceil$πm(m + 1) - π 2/4$\rceil$ ≤ N(k) ≤ $\lfloor$πm(m + 1) + 1$\rfloor$, where k (= 2m + 1) is any odd positive integer. We then extend this result to a more general family of point set, and give an O(hk log k) time algorithm for determining whether a given set of k points, having h points on the convex hull, belongs to such a family.