  In this paper we study a generalized version of the Weber problem of finding
a point that minimizes the sum of its distances to a finite number of given
points. In our setting these distances may be $cut$ $off$ at a given value $C >
0$, and we allow for the option of an $empty$ solution at a fixed cost $C'$. We
analyze under which circumstances these problems can be reduced to the simpler
Weber problem, and also when we definitely have to solve the more complex
problem with cutoff. We furthermore present adaptions of the algorithm of
[Drezner et al., 1991, $Transportation$ $Science$ 25(3), 183--187] to our
setting, which in certain situations are able to substantially reduce
computation times as demonstrated in a simulation study. The sensitivity with
respect to the cutoff value is also studied, which allows us to provide an
algorithm that efficiently solves the problem simultaneously for all $C>0$.
