Normal cycles and curvature measures of sets with d.c. boundary

TL;DR

This paper proves the existence and uniqueness of Legendrian cycles with curvature measures satisfying a local Gauss-Bonnet formula for sets with d.c. boundaries in any dimension, extending previous results to higher dimensions and more general sets.

Contribution

It establishes the existence and uniqueness of Legendrian cycles with curvature measures for d.c. boundary sets in all dimensions, including lower-dimensional sets, and describes their local index functions.

Findings

Existence and uniqueness of Legendrian cycles for d.c. boundary sets in all dimensions.

Curvature measures satisfy a local Gauss-Bonnet formula.

Curvature measures fulfill the Crofton formula.

Abstract

We show that for every compact domain in a Euclidean space with d.c. (delta-convex) boundary there exists a unique Legendrian cycle such that the associated curvature measures fulfil a local version of the Gauss-Bonnet formula. This was known in dimensions two and three and was open in higher dimensions. In fact, we show this property for a larger class of sets including also lower-dimensional sets. We also describe the local index function of the Legendrian cycles and we show that the associated curvature measures fulfill the Crofton formula.