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.

💡 Research Summary

The paper investigates the Fermat‑Weber point (the point minimizing the sum of Euclidean distances) for a very specific geometric configuration: a “k‑chain” formed by k consecutive vertices of a regular n‑gon. The authors distinguish two cases depending on whether k is odd or even and derive precise conditions under which the Fermat‑Weber point coincides with a distinguished vertex of the chain.

Main contributions

1. Odd‑length chains (k = 2m + 1).
   By placing the vertices on a unit circle centered at the origin and aligning the chain’s axis of symmetry with the x‑axis, the sum‑of‑distances function can be written as a convex function ψ(k,n,x) of a single variable x∈