Principal Kinematic Formulas for Germs of Closed Definable Sets

TL;DR

This paper establishes two principal kinematic formulas for germs of closed definable sets in Euclidean space, extending classical formulas by integrating over a specific manifold rather than Euclidean motions.

Contribution

It generalizes existing kinematic formulas to germs of closed definable sets, using integration over the manifold SO(n) × S^{n-1} instead of Euclidean motions.

Findings

Generalized Cauchy-Crofton formula for definable sets

Extended infinitesimal linear kinematic formula

Formulas applicable to germs of closed definable sets

Abstract

We prove two principal kinematic formulas for germs of closed definable sets in $\mathbb{R}^n$, that generalize the Cauchy-Crofton formula for the density due to Comte and the infinitesimal linear kinematic formula due to the author. In this setting, we do not integrate on the space of euclidian motions, but on the manifold $SO(n) \times S^{n-1}$.