Minsum Location Extended to Gauges and to Convex Sets

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 twofold way: more general than for normed spaces, the unit balls of our spaces are compact convex sets having the origin as an 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 corresponding theorems known for normed spaces. It turns out that some of these results holding for normed spaces carry over to our more general setting, and others do not. In addition, we present analogous results for related questions, like, e.g., for Heron’s problem. And finally, we derive a collection of additional results holding particularly for the Euclidean norm.