Curvature Measures of Singular Sets

  • Jan Rataj
  • Martina Zähle

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Jan Rataj, Martina Zähle
    Pages 1-36
  3. Jan Rataj, Martina Zähle
    Pages 37-45
  4. Jan Rataj, Martina Zähle
    Pages 47-53
  5. Jan Rataj, Martina Zähle
    Pages 55-86
  6. Jan Rataj, Martina Zähle
    Pages 87-103
  7. Jan Rataj, Martina Zähle
    Pages 105-138
  8. Jan Rataj, Martina Zähle
    Pages 139-158
  9. Jan Rataj, Martina Zähle
    Pages 159-170
  10. Jan Rataj, Martina Zähle
    Pages 171-207
  11. Jan Rataj, Martina Zähle
    Pages 209-244
  12. Back Matter
    Pages 245-256

About this book


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

Authors and affiliations

  • Jan Rataj
    • 1
  • Martina Zähle
    • 2
  1. 1.Mathematical InstituteCharles UniversityPragueCzech Republic
  2. 2.Mathematisches InstitutFriedrich-Schiller-Universität JenaJenaGermany

Bibliographic information

  • DOI
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-18182-6
  • Online ISBN 978-3-030-18183-3
  • Series Print ISSN 1439-7382
  • Series Online ISSN 2196-9922
  • Buy this book on publisher's site