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 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.Comment: submitted to Journal of Optimization Theory and Application

A generalization of the concept of distance based on the simplex inequality

“…The minimum in ( 14) is realized by any y 0 ∈ V that realizes the minimum of the values for y ∈ V . By definition, the vertex y 0 = m(u, v, w) is on shortest paths between any two elements among u, v, w, which shows that it realizes the minimum of each of the three terms in (15), and hence the minimum in (14). It follows that…”

“…We refer to [3,Chapter II] and [12] for an account of the history of this problem. Also, in [15], the location problem is extended in various directions and studied also for very general metrics -more general than those of normed spaces.…”

Extremal radii, diameter and minimum width in generalized Minkowski spaces

“…The increasing interest in vector spaces equipped with Minkowski functionals can be observed in various directions. For example, gauges or convex distance functions occur in computational geometry, operations research, and location science (see, e.g., [11,13,16,18,23,24]. In the present paper, a gentle start is provided for applying this setting to basic metrical notions of convex geometry.…”

Solving a Continuous Multifacility Location Problem by DC Algorithms

“…This problem and its extended version that involves a finite number of points in higher dimensions are examples of continuous single facility location problems. Over the years several generalized models of the Fermat-Torricelli type have been introduced and studied in the literature with practical applications to facility location decisions; see [8,12,14,16,18,19] and the references therein. An important feature of single facility location problems and the problems studied in the aforementioned references is that only one center/server has to be found to serve a finitely many demand points/customers.…”