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 fulfill 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. (C) 2013 Elsevier Inc.

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Keywords

• D.c. function
• Subgradient
• Monge-Ampere function
• Aura
• Curvature measure
• Legendrian cycle
• Gauss-Bonnet formula