  The curvature measures of a set $X$ with singularities are measures
concentrated on the normal bundle of $X$, which describe the local geometry of
the set $X$. For given finitely many convex bodies or, more generally, sets
with positive reach, the translative integral formula for curvature measures
relates the integral mean of the curvature measures of the intersections of the
given sets, one fixed and the others translated, to the mixed curvature
measures of the given sets. In the case of two sets of positive reach, a
representation of these mixed measures in terms of generalized curvatures,
defined on the normal bundles of the sets, is known. For more than two sets, a
description of mixed curvature measures in terms of rectifiable currents has
been derived previously. Here we provide a representation of mixed curvatures
measures of sets with positive reach based on generalized curvatures.
  The special case of convex polyhedra is treated in detail.
