Curvature Measures of Singular Sets

1. Front Matter

   Pages i-xi

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2. Background from Geometric Measure Theory

   • Jan Rataj, Martina Zähle

   Pages 1-36

3. Background from Convex Geometry

   • Jan Rataj, Martina Zähle

   Pages 37-45

4. Background from Differential Geometry and Topology

   • Jan Rataj, Martina Zähle

   Pages 47-53

5. Sets with Positive Reach

   • Jan Rataj, Martina Zähle

   Pages 55-86

6. Unions of Sets with Positive Reach

   • Jan Rataj, Martina Zähle

   Pages 87-103

7. Integral Geometric Formulas

   • Jan Rataj, Martina Zähle

   Pages 105-138

8. Approximation of Curvatures

   • Jan Rataj, Martina Zähle

   Pages 139-158

9. Characterization Theorems

   • Jan Rataj, Martina Zähle

   Pages 159-170

10. Extensions of Curvature Measures to Larger Set Classes

    • Jan Rataj, Martina Zähle

    Pages 171-207

11. Fractal Versions of Curvatures

    • Jan Rataj, Martina Zähle

    Pages 209-244

12. Back Matter

    Pages 245-256

    PDF

The book describes how curvature measures can be introduced for certain classes of sets with singularities in Euclidean spaces. Its focus lies on sets with positive reach and some extensions, which include the classical polyconvex sets and piecewise smooth submanifolds as special cases. The measures under consideration form a complete system of certain Euclidean invariants. Techniques of geometric measure theory, in particular, rectifiable currents are applied, and some important integral-geometric formulas are derived. Moreover, an approach to curvatures for a class of fractals is presented, which uses approximation by the rescaled curvature measures of small neighborhoods. The book collects results published during the last few decades in a nearly comprehensive way.

• Curvature Measure
• Gauss-Bonnet Theorem
• Geometric Measure Theory
• Principal Kinematic Formula
• Steiner Formula

“The presentation is clear and concise, and detailed proofs are given. … The book is certainly well suited for the serious student or researcher in another field who wants to learn the topic. … Students will learn the rigorous theoretical foundations for the subject as well as meet a large number of interesting examples.” (Lars Olsen, Mathematical Reviews, April, 2021)

“The monograph is well-written and the main concepts are clearly explained and presented. The contents are a comprehensive collection of results published during the last decades. The material is accessible to graduate students with a good background in geometric measure theory, convex analysis, and differential geometry. For researchers the volume is attractive for its overall point-of-view and the broad presentation of the subject.” (Peter Massopust, zbMATH 1423.28001, 2019)

• Mathematical Institute, Charles University, Prague, Czech Republic

  Jan Rataj

• Mathematisches Institut, Friedrich-Schiller-Universität Jena, Jena, Germany

  Martina Zähle

Jan Rataj, born in 1962 in Prague, studied at Charles University in Prague and defended his PhD at the Mathematical Institute of the Czech Academy of Sciences in 1991. He has been affiliated to Charles University in Prague since 1992, as full professor since 2000. He is the author of approximately 55 publications (on probability theory, stochastic geometry, mathematical analysis, differential and integral geometry).

Martina Zähle, born in1950, obtained her Diploma in 1973 from Moscow State University. She received a PhD in 1978 and Habilitation in 1982 from the Friedrich Schiller University Jena where she has also held the Chair of Probability Theory in 1988, and Geometry in 1991. She has co-edited the proceedings of the international conference series ‘’Fractal Geometry and Stochastics I -V’’, published by Birkhäuser and is the author of more than 100 publications (on geometric integration theory, fractal geometry, stochastic geometry, potential analysis, fractional calculus and (s)pde).

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