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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 ⌈πm(m + 1) - π2/4⌉ ≤ N(k) ≤ ⌊πm(m + 1) + 1⌋, 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.
This is a pre-publication version. The final publication is available at IOS Press through http://dx.doi.org/10.3233/FI-2011-406
computational geometry, facility location, Fermat-Weber problem, optimization, polygons
Bhattacharya, B. B. (2011). On the Fermat-Weber Point of a Polygonal Chain and Its Generalizations. Fundamenta Informaticae, 107 (4), 331-343. http://dx.doi.org/10.3233/FI-2011-406
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Date Posted: 25 October 2018
This document has been peer reviewed.