The Fermat-Torricelli problem in the case of three-point sets in normed planes

TL;DR

This paper investigates the uniqueness of solutions to the Fermat-Torricelli problem for three points in normed planes, providing a criterion for uniqueness and analyzing specific norms like lambda planes.

Contribution

It introduces a criterion for solution uniqueness in normed planes and applies it to norms defined by regular polygons, advancing understanding of the problem's geometric properties.

Findings

Established a criterion for solution uniqueness in normed planes.

Applied the criterion to lambda planes defined by regular polygons.

Identified conditions under which the Fermat-Torricelli problem has unique solutions.

Abstract

In the paper the Fermat-Torricelli problem is considered. The problem asks a point minimizing the sum of distances to arbitrarily given points in d-dimensional real normed spaces. Various generalizations of this problem are outlined, current methods of solving and some recent results in this area are presented. The aim of the article is to find an answer to the following question: in what norms on the plane is the solution of the Fermat-Torricelli problem unique for any three points. The uniqueness criterion is formulated and proved in the work, in addition, the application of the criterion on the norms set by regular polygons, the so-called lambda planes, is shown.