On the Geometric Median of Convex, Triangular and Other Polygonal Domains

• Main Author: P. A. Panov
• Format: Article
• Language: English
• Published: Irkutsk State University 2018-12-01
• Series: Известия Иркутского государственного университета: Серия "Математика"
• Subjects: geometric median; location problem; convex domain; distance to the boundary; gradient system
• Online Access: http://mathizv.isu.ru/en/article/file?id=1282

The classical Fermat-Torricelli problem consists in finding the point which minimizes the sum of distances from it to the three vertices of a given triangle. This problem has various generalizations. For example, given a subset $S$ of the plane consisting of $n$ points, one can look for a point that minimizes the sum of $n$ distances, i.e., the median of $S$. A similar question can be asked for a Euclidean space of any dimension or for any metric space. The generalized Fermat-Torricelli problem concerns minimizing a weighted sum of distances, and it is one of the main problems in Facility Location theory. An analytic solution of Fermat-Torricelli problem is non-trivial even in the case of three points, and the general case is quite complex. In this work we consider a further generalization, namely the continuous case in which we look for a geometric median of a two-dimensional domain, where the sum of distances is being replaced by an integral. It is rather straightforward to see that the median of a convex domain $\Omega$ is contained in its interior. In this article we find a universal geometric bound for the distance from the median to the boundary of $\Omega$, which only depends on the area, $S(\Omega)$, and its diameter $d(\Omega)$. Also, we look into polygonal domains. Even in the case of a triangular domain, one can hardly expect an explicit analytic (closed-form) solution. However, using elementary functions, one can obtain a gradient system for finding the geometric median of a triangular domain. By using a triangulation of a polygonal domain, this result can be generalized to polygonal domains. In addition, we discuss in detail the geometric properties of isosceles triangles.