  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.
