The Fermat-Torricelli problem in normed planes and spaces

TL;DR

This paper explores the Fermat-Torricelli problem in two-dimensional and higher-dimensional normed spaces, providing new geometric insights and consolidating existing scattered results into a comprehensive minitheory.

Contribution

It introduces a geometric approach to analyze the Fermat-Torricelli locus in Minkowski spaces, presenting new results and unifying known findings.

Findings

New geometric characterizations of the Fermat-Torricelli locus

A collection of scattered results unified into a minitheory

Demonstration of geometric methods' effectiveness in Minkowski spaces

Abstract

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.