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Integral Geometric Regularity

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Tensor Valuations and Their Applications in Stochastic Geometry and Imaging

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2177))

Abstract

Smooth scalar-valued valuations may be thought of as curvature integrals that are robust enough to apply to objects with convex singularities. It turns out that certain kinds of nonconvex singularities are also included. The distinguishing feature is the existence of a normal cycle, which is an integral current that stands in for the manifold of unit normals in case they do not exist in the usual sense. We describe the elements of the normal cycle construction, and sketch how it may be used to establish the fundamental relations of integral geometry, with emphasis on the class of WDC sets recently introduced by Pokorný and Rataj.

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Acknowledgements

Partially supported by NSF grant DMS-1406252. Many thanks to the anonymous referee for a number of helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Joseph H. G. Fu

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  1. Joseph H. G. Fu
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Correspondence to Joseph H. G. Fu .

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Editors and Affiliations

  1. Department of Mathematics, Aarhus University, Aarhus C, Denmark

    Eva B. Vedel Jensen

  2. Department of Mathematics, Aarhus University, Aarhus C, Denmark

    Markus Kiderlen

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Fu, J.H.G. (2017). Integral Geometric Regularity. In: Jensen, E., Kiderlen, M. (eds) Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol 2177. Springer, Cham. https://doi.org/10.1007/978-3-319-51951-7_10

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