Lipschitz-Killing curvatures and polar images

Abstract

We relate the Lipschitz-Killing measures of a definable set $X\subset {\mathbb{R}}^n$ in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of ${\mathbb{R}}^n$, such results were established by Langevin and Shifrin.Then we give infinitesimal versions of these results. As a corollary, we obtain a relation between the polar invariants of Comte and Merle and the densities of generic polar images.