Minsum Location Extended to Gauges and to Convex Sets

TL;DR

This paper generalizes the Fermat-Torricelli problem to settings with convex, possibly asymmetric unit balls and convex sets, extending classical results from normed spaces to more general convex geometric contexts.

Contribution

It introduces a two-fold extension of the Fermat-Torricelli problem to convex sets with non-symmetric unit balls and explores related geometric optimization problems.

Findings

Some classical theorems extend to the new setting.

New theorems are established for convex sets and asymmetric norms.

Results include special cases for Euclidean norms.

Abstract

One of the oldest and richest problems from continuous location science is the famous Fermat-Torricelli problem, asking for the unique point in Euclidean space that has minimal distance sum to n given (non-collinear) points. Many natural and interesting generalizations of this problem were investigated, e.g., by extending it to non-Euclidean spaces and modifying the used distance functions, or by generalizing the configuration of participating geometric objects. In the present paper, we extend the Fermat-Torricelli problem in a two-fold way: more general than for normed spaces, the unit balls of our spaces are compact convex sets having the origin as interior point (but without symmetry condition), and the n given objects can be general convex sets (instead of points). We combine these two viewpoints, and the presented sequence of new theorems follows in a comparing sense that of theorems known for normed spaces. Some of these results holding for normed spaces carry over to our more general setting, and others not. In addition, we present analogous results for related questions, like, e.g., for Heron's problem. And finally we derive a collection of results holding particularly for the Euclidean norm.