On the Fermat-Weber Point of a Polygonal Chain

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 $k(\leq n)$ 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 $n\geq N(k)$. We also show that $\lceil\pi m(m+1)-\pi^2/4\rceil \leq N(k) \leq \lfloor\pi m(m+1)+1\rfloor$, 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.