The Fermat-Torricelli problem in normed planes and spaces

We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach.

💡 Research Summary

The paper investigates the classical Fermat‑Torricelli (FT) problem—minimising the sum of distances from a variable point to a finite set of given points—in the setting of real normed vector spaces, with a particular focus on two‑dimensional Minkowski planes. After a brief historical introduction, the authors lay out the necessary preliminaries: a normed space ((\mathbb R^{d},|\cdot|)), the definition of an FT point as a minimiser of (F(x)=\sum_{i=1}^{n}|x-a_{i}|), and the FT locus as the set of all such minimisers. They prove that the FT locus is always a non‑empty, closed, convex set, regardless of the norm.

The core of the work is a geometric analysis based on duality and support functions. For each fixed site (a_{i}), the subgradient of the distance function (|x-a_{i}|) is identified with the set of outward normals to the unit ball at the point where the ray from (a_{i}) through (x) meets the boundary. Consequently, a point (x) is an FT point precisely when the sum of these subgradients contains the zero vector. This condition translates into an intersection of supporting half‑spaces, giving an explicit description of the FT locus in terms of the geometry of the unit ball.

Two major families of results are obtained. First, when the norm is strictly convex (the unit ball has no flat faces), the FT locus collapses to a single point; thus uniqueness of the FT point holds for all configurations, mirroring the Euclidean case. Second, when the norm is not smooth—e.g., the unit ball is a polygon—the FT locus can be a line segment, a polygon, or a higher‑dimensional polytope. In the planar case, the authors show that the FT locus coincides with the central polygon associated with the unit ball: for the (\ell_{1}) norm (diamond‑shaped unit ball) the FT locus is the axis‑aligned square formed by the medians of the data points; for the (\ell_{\infty}) norm (square unit ball) it becomes an axis‑parallel square; for a regular hexagonal norm it is a smaller hexagon centred at the same point. These concrete examples illustrate how the symmetry and number of faces of the unit ball directly dictate the shape of the FT locus.

Beyond existence and shape, the paper establishes several stability properties. The FT locus varies continuously (in the Hausdorff metric) under small perturbations of the norm, and the authors provide quantitative bounds on how much the locus can move when the norm is slightly altered. They also prove a “minitheory” that gathers the fundamental facts—existence, convexity, uniqueness under strict convexity, continuity, and symmetry—into a compact framework that can be used as a reference for future work on location problems in arbitrary normed spaces.

The manuscript concludes with a discussion of open problems, notably the description of FT loci in higher dimensions for non‑smooth norms and the extension of the geometric approach to asymmetric (non‑centrally symmetric) norms. Overall, the paper demonstrates that a purely geometric viewpoint yields powerful, norm‑independent insights into the Fermat‑Torricelli problem and provides a solid foundation for algorithmic and theoretical developments in facility location, network design, and related optimisation fields.