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

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.

22 pp, corrected version

References in corpus (1)

• On compositions of d.c. functions and mappings

Cited by in corpus (4)

• On the structure of sets with positive reach
• Valuation theory of indefinite orthogonal groups
• Hölder continuity of normal cycles and of support measures of convex bodies
• Legendrian cycles and curvatures