The Fermat-Torricelli Problem and Weiszfeld's Algorithm in the Light of Convex Analysis

TL;DR

This paper revisits the classical Fermat-Torricelli problem, which seeks a point minimizing total distance to given points, using convex analysis and optimization techniques for theoretical insights and numerical solutions.

Contribution

It provides a new convex analysis framework for the Fermat-Torricelli problem and explores numerical methods for solving it in higher dimensions.

Findings

Convex analysis offers a unified approach to the problem.

Numerical algorithms are developed for efficient solutions.

Theoretical insights improve understanding of solution properties.

Abstract

In the early 17th century, Pierre de Fermat proposed the following problem: given three points in the plane, find a point such that the sum of its Euclidean distances to the three given points is minimal. This problem was solved by Evangelista Torricelli and was named the {\em Fermat-Torricelli problem}. A more general version of the Fermat-Torricelli problem asks for a point that minimizes the sum of the distances to a finite number of given points in $\Bbb R^n$. This is one of the main problems in location science. In this paper we revisit the Fermat-Torricelli problem from both theoretical and numerical viewpoints using some ingredients of convex analysis and optimization.