Boris Mordukhovich, Nguyen Mau Nam
Published 2013-02-21, updated 2019-12-23Version 4
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.
Some remarks on convex analysis in topological groups
Convex Analysis at Infinity: An Introduction to Astral Space
Explicit formulas for projections onto intersection of a finite number of halfspaces
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