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

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.

preprint2013arXivOpen access
Dusan PokornyJan Rataj
math.DG
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Dusan PokornyJan Rataj

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