  The Fermat-Weber location problem requires finding a point in $\mathbb{R}^n$
that minimizes the sum of weighted Euclidean distances to $m$ given points. An
iterative solution method for this problem was first introduced by E. Weiszfeld
in 1937. Global convergence of Weiszfeld's algorithm was proven by W. Kuhn in
1973. This paper studies Fermat-Weber location problems with closed convex
constraint sets in real Hilbert spaces. In section two, we show that existence
and uniqueness of solutions of the problems. Moreover, the solution is stable
with respect to the perturbation of the $m$ anchor points. In the section
three, we extend Weiszfeld's algorithm by adding a projection on the constraint
set. The convergence of the sequence generated by the method to the optimal
solution of the problem is proved.
