Location of a moving service facility

In this paper we consider the general question in the field of mathematics of whether some properties or algorithms that hold in finite dimension spaces also hold in function spaces. We answer this question concerning the very well-known Weiszfeld algorithm for the Weber problem. In order to do that, we consider the Weber problem with trajectories (functions of time) instead of points in a finite-dimensional space. This is in fact the problem of locating a moving service facility. Properties are proved assuring that the problem is well-established and that an optimal solution exists if L^p 1 ≤ p ≤ +∞ spaces are considered. An extension of Weiszfeld's algorithm is proposed to solve this kind of problem and it is shown that under some assumptions it presents global convergence properties. Moreover, an example is included showing that this extension is not trivial because the natural pointwise extension of Weiszfeld's algorithm does not have to converge to an optimal solution of the considered problem while the new algorithm does.