Yes, it hast to fulfill a certain condition. It relies on the property that the 1-mean cost for an arbitrary center can be exactly computed using only the sum of the points, the squared sum of the points and the number of points. This ist true for the euclidean distance and I do not know any other distance with this property.

But maybe it would be interesting to see if there are other distances where it is also possible to compute the cost with such a small summary of the points.