Advertisement

Crofton Formulae for Tensorial Curvature Measures: The General Case

  • Daniel Hug
  • Jan A. Weis
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 25)

Abstract

The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. On convex polytopes, there exist further generalizations some of which also have continuous extensions to arbitrary convex bodies. In a previous work, we obtained kinematic formulae for all (generalized) tensorial curvature measures. As a consequence of these results, we now derive a complete system of Crofton formulae for such (generalized) tensorial curvature measures. These formulae express the integral mean of the (generalized) tensorial curvature measures of the intersection of a given convex body (resp. polytope, or finite unions thereof) with a uniform affine k-flat in terms of linear combinations of (generalized) tensorial curvature measures of the given convex body (resp. polytope, or finite unions thereof). The considered generalized tensorial curvature measures generalize those studied formerly in the context of Crofton-type formulae, and the coefficients involved in these results are substantially less technical and structurally more transparent than in previous works. Finally, we prove that essentially all generalized tensorial curvature measures on convex polytopes are linearly independent. In particular, this implies that the Crofton formulae which we prove in this contribution cannot be simplified further.

Notes

Acknowledgements

The authors were supported in part by DFG grants FOR 1548.

References

  1. 1.
    S. Alesker, Description of continuous isometry covariant valuations on convex sets. Geom. Dedicata 74, 241–248 (1999)Google Scholar
  2. 2.
    E. Artin, The Gamma Function (Holt, Rinehart and Winston, New York, 1964)Google Scholar
  3. 3.
    A. Bernig, D. Hug, Kinematic formulas for tensor valuations. J. Reine Angew. Math. (2015). arXiv:1402.2750v2. https://doi.org/10.1515/crelle-2015-002Google Scholar
  4. 4.
    H. Federer, Curvature measures. Trans. Amer. Math. Soc. 93, 418–491 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. Glasauer, Integralgeometrie konvexer Körper im sphärischen Raum. Dissertation. Albert-Ludwigs-Universität Freiburg, Freiburg (1995)zbMATHGoogle Scholar
  6. 6.
    S. Glasauer, A generalization of intersection formulae of integral geometry. Geom. Dedicata 68, 101–121 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. Hadwiger, Additive Funktionale k-dimensionaler Eikörper I. Arch. Math. 3, 470–478 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    H. Hadwiger, R. Schneider, Vektorielle Integralgeometrie. Elem. Math. 26, 49–57 (1971)Google Scholar
  9. 9.
    D. Hug, R. Schneider, Local tensor valuations. Geom. Funct. Anal. 24, 1516–1564 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Hug, R. Schneider, SO(n) covariant local tensor valuations on polytopes. Mich. Math. J. 66, 637–659 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Hug, R. Schneider, Rotation covariant local tensor valuations on convex bodies. Commun. Contemp. Math. 19, 1650061, 31 pp. (2017). https://doi.org/10.1142/S0219199716500619
  12. 12.
    D. Hug, J.A. Weis, Kinematic formulae for tensorial curvature measures. Ann. Mat. Pura Appl. arXiv: 1612.08427 (2016)Google Scholar
  13. 13.
    D. Hug, J.A. Weis, Crofton formulae for tensor-valued curvature measures, in Tensor Valuations and their Applications in Stochastic Geometry and Imaging, ed. by M. Kiderlen, E.B. Vedel Jensen. Lecture Notes in Mathematics, vol. 2177 (Springer, Berlin, 2017), pp. 111–156. https://doi.org/10.1007/978-3-319-51951-75
  14. 14.
    D. Hug, J.A. Weis, Integral formulae for Minkowski tensors. arXiv: 1712.09699 (2017)Google Scholar
  15. 15.
    D. Hug, R. Schneider, R. Schuster, Integral geometry of tensor valuations. Adv. Appl. Math. 41, 482–509 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    D. Hug, R. Schneider, R. Schuster, The space of isometry covariant tensor valuations. Algebra i Analiz 19, 194–224 (2007); St. Petersburg Math. J. 19, 137–158 (2008)Google Scholar
  17. 17.
    M. Kiderlen, E.B. Vedel Jensen, Tensor Valuations and their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177 (Springer, Berlin, 2017)Google Scholar
  18. 18.
    P. McMullen, Isometry covariant valuations on convex bodies. Rend. Circ. Mat. Palermo (2), Suppl. 50, 259–271 (1997)Google Scholar
  19. 19.
    K.R. Mecke, Additivity, convexity, and beyond: applications of Minkowski functionals in statistical physics, in Statistical Physics and Spatial Statistics, ed. by K.R. Mecke, D. Stoyan. Lecture Notes in Physics, vol. 554 (Springer, Berlin, 2000)Google Scholar
  20. 20.
    M. Saienko, Tensor-valued valuations and curvature measures in Euclidean spaces. PhD Thesis, University of Frankfurt (2016)Google Scholar
  21. 21.
    R. Schneider, Krümmungsschwerpunkte konvexer Körper. I. Abh. Math. Sem. Univ. Hamburg 37, 112–132 (1972)CrossRefzbMATHGoogle Scholar
  22. 22.
    R. Schneider, Krümmungsschwerpunkte konvexer Körper. II. Abh. Math. Sem. Univ. Hamburg 37, 204–217 (1972)CrossRefzbMATHGoogle Scholar
  23. 23.
    R. Schneider, Kinematische Berührmaße für konvexe Körper. Abh. Math. Sem. Univ. Hamburg 44, 12–23 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    R. Schneider, Curvature measures of convex bodies. Ann. Mat. Pura Appl. 116, 101–134 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    R. Schneider, Local tensor valuations on convex polytopes. Monatsh. Math. 171, 459–479 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151 (Cambridge University Press, Cambridge, 2014)Google Scholar
  27. 27.
    R. Schneider, W. Weil, Stochastic and Integral Geometry (Springer, Berlin, 2008)CrossRefzbMATHGoogle Scholar
  28. 28.
    G.E. Schröder-Turk et al., Minkowski tensor shape analysis of cellular, granular and porous structures. Adv. Mat. 23, 2535–2553 (2011)CrossRefGoogle Scholar
  29. 29.
    A.M. Svane, E.B. Vedel Jensen, Rotational Crofton formulae for Minkowski tensors and some affine counterparts. Adv. Appl. Math. 91, 44–75 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations