Lipschitz-Killing curvatures and polar images

TL;DR

This paper establishes a relationship between Lipschitz-Killing measures of definable sets and the volumes of their polar images, extending known results for smooth manifolds to more general sets and providing infinitesimal versions.

Contribution

It generalizes the connection between Lipschitz-Killing measures and polar images from smooth manifolds to definable sets in o-minimal structures, including infinitesimal versions.

Findings

Relates Lipschitz-Killing measures to volumes of polar images.

Provides infinitesimal versions of these geometric relations.

Connects polar invariants with densities of 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.