Median problems in location science

Abstract

In this chapter we are concerned with problems of the following type: Given a finite set of weighted points in Euclidean n-space, n ≥ 2, we are interested in the location of an additonal point such that the sum of weighted distances to the given points is minimal. Historically correct, this is the (generalized) Fermat-Torricelli problem, and in location science it is also called the Steiner- Weber problem and the 1-median problem, respectively. It is our aim to give a complete mathematical approach to this problem, to present several historical corrections and to study various natural generalizations of it. These generalizations refer to the distance measure under consideration (namely, by extending the problem to finite-dimensional normed spaces) and to the geometric configuration itself (replacement of the searched optimal point by a hyperplane, or by a flat of some other dimension). In all these cases, we first establish some necessary position criteria with the help of which then algorithmical approaches are obtained. So this chapter might be interesting from the viewpoint of classical and computational geometry, location science, and other applied disciplines (such as, e.g., robust statistics).