Titlebook: Curvature Measures of Singular Sets; Jan Rataj,Martina Zähle Book 2019 Springer Nature Switzerland AG 2019 Curvature Measure.Gauss-Bonnet

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https://doi.org/10.1057/978-1-137-54060-7In this chapter we will give a brief introduction to certain fractal versions of Lipschitz-Killing curvature measures. We present only some of the proofs giving insight into certain techniques which are applied in this and different fields. For the other proofs we refer to the literature.

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fiscal

Sets with Positive Reach,The metric projection to a set . is not determined everywhere unless . is convex and closed. Sets with positive reach are sets with the property that the metric projection is defined on some open neighbourhood of ..

ATRIA

Fractal Versions of Curvatures,In this chapter we will give a brief introduction to certain fractal versions of Lipschitz-Killing curvature measures. We present only some of the proofs giving insight into certain techniques which are applied in this and different fields. For the other proofs we refer to the literature.

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Background from Geometric Measure Theory, and results from this chapter can be found in the Federer’s book [.] and/or in the book of Krantz and Parks [.]. Other sources will be cited when needed. Most of these results are presented without proofs.

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Characterization Theorems,ir geometric importance. Recall that a well-known result of Hadwiger (Theorem .) states that any motion invariant continuous valuation on the space of compact convex sets is a linear combination of Minkowski’s quermasssintegrals.

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Extensions of Curvature Measures to Larger Set Classes,ive reach). This setting is still not satisfactory since it does not encompass some natural set classes as closures of complements to convex bodies, or boundaries of convex bodies, though these sets should apparently admit a natural definition of curvatures.

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ULCER