  The continuity of the inverse Klain map is investigated and the class of
centrally symmetric convex bodies at which every valuation depends continuously
on its Klain function is characterized. Among several applications, it is shown
that McMullen's decomposition is not possible in the class of
translation-invariant, continuous, positive valuations. This implies that there
exists no McMullen decomposition for translation-invariant, continuous
Minkowski valuations, which solves a problem first posed by Schneider and
Schuster.
