Differentiability of the volume of a region enclosed by level sets

The level of a function f on Rn encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f . For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important