Kinematic formulas for sets defined by differences of convex functions

TL;DR

This paper extends kinematic formulas to sets defined by differences of convex functions on manifolds, generalizing Federer's theory and providing new curvature measure invariants for these sets.

Contribution

It introduces invariant curvature measures for WDC sets on manifolds and establishes kinematic formulas applicable to these non-smooth sets, extending classical convex geometry results.

Findings

Defined invariant curvature measures for WDC sets

Proved kinematic formulas for these sets under group actions

Extended Federer's theory of sets with positive reach

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.