Kinematic formulas for sets defined by differences of convex functions

Abstract

Two of the authors have defined the class $ WDC(M)$ as the class of all subsets of a smooth manifold $M$ that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If $M$ is Riemanian and $G$ is a group of isometries acting transitively on the sphere bundle $SM$, we define the invariant curvature measures of compact \WDC~ subsets of $M$, and show that pairs of such subsets are subject to the array of kinematic formulas known to apply to smoother sets. Restricting to the case $(M, G) = (\mathbb R^d, \overline{SO(d)})$, this extends and subsumes Federer's theory of sets with positive reach in an essential way. The key technical point is equivalent to a sharpening of a classical theorem of Ewald, Larman, and Rogers characterizing the dimension of the set of directions of line segments lying in the boundary of a given convex body.