Advertisement

Annali di Matematica Pura ed Applicata (1923 -)

, Volume 197, Issue 5, pp 1349–1384 | Cite as

Kinematic formulae for tensorial curvature measures

  • Daniel HugEmail author
  • Jan A. Weis
Article
  • 81 Downloads

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. We prove a complete set of kinematic formulae for such (generalized) tensorial curvature measures. These formulae express the integral mean of the (generalized) tensorial curvature measure of the intersection of two given convex bodies (resp. polytopes), one of which is uniformly moved by a proper rigid motion, in terms of linear combinations of (generalized) tensorial curvature measures of the given convex bodies (resp. polytopes). We prove these results in a more direct way than in the classical proof of the principal kinematic formula for curvature measures, which uses the connection to Crofton formulae to determine the involved constants explicitly.

Keywords

Kinematic formula Tensor valuation Curvature measure Minkowski tensor Integral geometry Convex body Polytope 

Mathematics Subject Classification

Primary 52A20 53C65 Secondary 52A22 52A38 28A75 

References

  1. 1.
    Alesker, S.: Continuous rotation invariant valuations on convex sets. Ann. Math. 149, 977–1005 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alesker, S.: Description of continuous isometry covariant valuations on convex sets. Geom. Dedicata 74, 241–248 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alesker, S.: New structures on valuations and applications. In: Gallego, E., Solanes, G. (eds.) Integral Geometry and Valuations. Advanced Courses in Mathematics CRM Barcelona. Springer, Basel (2014)CrossRefGoogle Scholar
  4. 4.
    Alesker, S., Faifman, D.: Convex valuations in variant under the Lorentz group. J. Differ. Geom. 98, 183–236 (2014)CrossRefzbMATHGoogle Scholar
  5. 5.
    Artin, E.: The Gamma Function. Holt, Rinehart and Winston, New York (1964)zbMATHGoogle Scholar
  6. 6.
    Auneau-Cognacq, J., Ziegel, J., Vedel Jensen, E.B.: Rotational integral geometry of tensor valuations. Adv. Appl. Math. 50, 429–444 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Auneau, J., Rataj, J., Vedel Jensen, E.B.: Closed form of the rotational Crofton formula. Math. Nachr. 285, 164–180 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bernig, A.: Integral geometry under G2 and Spin(7). Isr. J. Math. 184, 301–316 (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bernig, A.: Invariant valuations on quaternionic vector spaces. J. Inst. Math. Jussieu 11, 467–499 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bernig, A.: Algebraic integral geometry. In: Bär, C., Lohkamp, J., Schwarz, M. (eds.) Global Differential Geometry. Springer Proceedings in Mathematics, vol. 17. Springer, Berlin (2012)CrossRefGoogle Scholar
  11. 11.
    Bernig, A.: Valuations and curvature measures on complex spaces. In: Kiderlen, M., Vedel Jensen, E.B. (eds.) Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177. Springer, Berlin (2017)Google Scholar
  12. 12.
    Bernig, A., Faifman, D.: Valuation theory of indefinite orthogonal groups. J. Funct. Anal. 273, 2167–2247 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bernig, A., Fu, J.H.G.: Hermitian integral geometry. Ann. Math. 2(173), 907–945 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bernig, A., Fu, J.H.G., Solanes, G.: Integral geometry of complex space forms. Geom. Funct. Anal. 24, 403–492 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bernig, A., Hug, D.: Kinematic formulas for tensor valuations. J. Reine Angew. Math. (2017+). https://doi.org/10.1515/crelle-2015-002. arXiv:1402.2750v2 (2015)
  16. 16.
    Bernig, A., Hug, D.: Integral geometry and algebraic structures for tensor valuations. In: Kiderlen, M., Vedel Jensen, E.B. (eds.) Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177. Springer, Berlin (2017)Google Scholar
  17. 17.
    Bernig, A., Solanes, G.: Classification of invariant valuations on the quaternionic plane. J. Funct. Anal. 267, 2933–2961 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bernig, A., Solanes, G.: Kinematic formulas on the quaternionic plane. Proc. Lond. Math. Soc. 3, 1–38 (2017)zbMATHGoogle Scholar
  19. 19.
    Bernig, A., Voide, F.: Spin-invariant valuations on the octonionic plane. Isr. J. Math. 214, 831–855 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Böbel, A., Räth, C.: Kinetics of fluid demixing in complex plasmas: domain growth analysis using Minkowski tensors. Phys. Rev. E 94, 013201 (2016).  https://doi.org/10.1103/PhysRevE.94.013201 CrossRefGoogle Scholar
  21. 21.
    Christensen, S.T., Kiderlen, M.: Comparison of two global digital algorithms for Minkowski tensor estimation. Centre for Stochastic Geometry and Advanced Bioimaging, research report (2016). http://pure.au.dk/portal/files/104174606/math_csgb_201610.pdf
  22. 22.
    Colesanti, A., Hug, D.: Hessian measures of semi-convex functions and applications to support measures of convex bodies. Manuscr. Math. 101, 209–238 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Faifman, D.: Crofton formulas and indefinite signature. Geom. Funct. Anal. 27, 489–540 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Fu, J.H.G.: Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39, 1115–1154 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fu, J.H.G.: Structure of the unitary valuation algebra. J. Differ. Geom. 72, 509–533 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Fu, J.H.G.: Algebraic integral geometry. In: Gallego, E., Solanes, G. (eds.) Integral Geometry and Valuations. Advanced Courses in Mathematics CRM Barcelona. Springer, Basel (2014)Google Scholar
  28. 28.
    Glasauer, S.: A generalization of intersection formulae of integral geometry. Geom. Dedicata 68, 101–121 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Goodey, P., Hug, D., Weil, W.: Kinematic formulas for area measures. Indiana Univ. Math. J. 66, 997–1018 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Goodey, P., Weil, W.: Translative and kinematic integral formulae for support functions. II. Geom. Dedicata 99, 103–125 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Goodey, P., Weil, W.: Sums of sections, surface area measures, and the general Minkowski problem. J. Differ. Geom. 97, 477–514 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hadwiger, H., Schneider, R.: Vektorielle Integralgeometrie. Elem. Math. 26, 49–57 (1971)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hörrmann, J.: The method of densities for non-isotropic Boolean models. Ph.D. Thesis, Karlsruhe Institute of Technology, KIT Scientific Publishing, Karlsruhe (2015). https://doi.org/10.5445/KSP/1000045101
  34. 34.
    Hörrmann, J., Hug, D., Klatt, M., Mecke, K.: Minkowski tensor density formulas for Boolean models. Adv. Appl. Math. 55, 48–85 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hug, D., Kiderlen, M., Svane, A.M.: Voronoi-based estimation of Minkowski tensors from finite point samples. Discrete Comput. Geom. 57, 545–570 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hug, D., Rataj, J.: Mixed curvature measures of translative integral geometry. Geom. Dedicata (2016).  https://doi.org/10.1007/s10711-017-0278-1 zbMATHGoogle Scholar
  37. 37.
    Hug, D., Schneider, R.: Local tensor valuations. Geom. Funct. Anal. 24, 1516–1564 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hug, D., Schneider, R.: SO\((n)\) covariant local tensor valuations on polytopes. Mich. Math. J. 66, 637–659 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hug, D., Schneider, R.: Rotation covariant local tensor valuations on convex bodies. Commun. Contemp. Math. 19, 1650061 (2017).  https://doi.org/10.1142/S0219199716500619 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Hug, D., Schneider, R.: Tensor valuations and their local versions. In: Kiderlen, M., Vedel Jensen, E.B. (eds.) Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177. Springer, Berlin (2017)Google Scholar
  41. 41.
    Hug, D., Schneider, R., Schuster, R.: Integral geometry of tensor valuations. Adv. Appl. Math. 41, 482–509 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Hug, D., Schneider, R., Schuster, R.: The space of isometry covariant tensor valuations. Algebra i Analiz 19, 194–224 (2007), St. Petersburg Math. J. 19, 137–158 (2008)Google Scholar
  43. 43.
    Hug, D., Weis, J.A.: Crofton formulae for tensor-valued curvature measures. In: Kiderlen, M., Vedel Jensen, E.B. (eds.) Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177. Springer, Berlin (2017)Google Scholar
  44. 44.
    Hug, D., Weis, J.A.: Kinematic formulae for tensorial curvature measures (2016). arXiv:1612.08427v1
  45. 45.
    Hug, D., Weis, J.A.: Crofton formulae for tensorial curvature measures: the general case. In: Bianchi, G., Colesanti, A., Gronchi, P. (eds.) Analytic Aspects of Convexity, Springer INdAM Series. Springer (2017+) (to appear). arXiv:1606.05131 (2016)
  46. 46.
    Hug, D., Weis, J.A.: Integral geometric formulae for Minkowski tensors (2017). ArXiv:1712.09699
  47. 47.
    Kapfer, S.C., Mickel, W., Mecke, K., Schröder-Turk, G.E.: Jammed spheres: Minkowski tensors reveal onset of local crystallinity. Phys. Rev. E 85, 030301-1–030301-4 (2012).  https://doi.org/10.1103/PhysRevE.85.030301 CrossRefGoogle Scholar
  48. 48.
    Kiderlen, M., Vedel Jensen, E.B.: Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177. Springer, Berlin (2017)zbMATHGoogle Scholar
  49. 49.
    Klatt, M.A.: Morphometry of random spatial structures in physics. Ph.D. Thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, FAU University Press (2016). https://opus4.kobv.de/opus4-fau/frontdoor/index/index/docId/7654
  50. 50.
    Kousholt, A.: Reconstruction of \(n\)-dimensional convex bodies from surface tensors. Centre for Stochastic Geometry and Advanced Bioimaging, research report (2016). http://pure.au.dk/portal/files/101273868/math_csgb_201608.pdf
  51. 51.
    Kousholt, A.: Minkowski tensors. Stereological estimation, reconstruction and stability results. Ph.D. Thesis, Aarhus University, Aarhus (2016). http://pure.au.dk/portal/files/103337616/math_phd_2016_ak.pdf
  52. 52.
    Kousholt, A., Kiderlen, M.: Reconstruction of convex bodies from surface tensors. Adv. Appl. Math 76, 1–33 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Kousholt, A., Kiderlen, M., Hug, D.: Surface tensor estimation from linear sections. Math. Nachr. 288, 1647–1672 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Kousholt, A., Ziegel, J.F., Kiderlen, M., Vedel Jensen, E.B.: Stereological estimation of mean particle volume tensors in \(\mathbb{R}^3\) from vertical sections. Centre for Stochastic Geometry and Advanced Bioimaging, research report (2016). http://pure.au.dk/portal/files/103277013/math csgb_2016_09.pdf
  55. 55.
    Kuhn, M.R., Sun, W., Wang, Q.: Stress-induced anisotropy in granular materials: fabric, stiffness, and permeability. Acta Geotech 10, 399–419 (2015)CrossRefGoogle Scholar
  56. 56.
    McMullen, P.: Isometry covariant valuations on convex bodies. Rend. Circ. Mat. Palermo 2(Suppl. 50), 259–271 (1997)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Rafati, A.H., Ziegel, J.F., Nyengaard, J.R., Vedel Jensen, E.B.: Stereological estimation of particle shape and orientation from volume tensors. J. Microsc. 261, 229–237 (2016)CrossRefGoogle Scholar
  58. 58.
    Saadatfar, M., Mukherjee, M., Madadi, M., Schröder-Turk, G.E., Garcia-Moreno, F., Schaller, F.M., Hutzler, S., Sheppard, A.P., Banhart, J., Ramamurty, U.: Structure and deformation correlation of closed-cell aluminium foam subject to uniaxial compression. Acta Mater. 60, 3604–15 (2012)CrossRefGoogle Scholar
  59. 59.
    Saienko, M.: Tensor-valued valuations and curvature measures in Euclidean spaces. Ph.D. Thesis, Goethe-Universität Frankfurt, Frankfurt am Main (2016). http://publikationen.ub.uni-frankfurt.de/files/42032/Saienko.pdf
  60. 60.
    Schaller, F.M., Kapfer, S.C., Hilton, J.E., Cleary, P.W., Mecke, K., De Michele, C., Schilling, T., Saadatfar, M., Schröter, M., Delaney, G.W., Schröder-Turk, G.E.: Non-universal Voronoi cell shapes in amorphous ellipsoid packs. Europhys. Lett. 111(2), 24002 (2015). http://stacks.iop.org/0295-5075/111/i=2/a=24002
  61. 61.
    Schneider, R.: Krümmungsschwerpunkte konvexer Körper. I. Abh. Math. Sem. Univ. Hambg. 37, 112–132 (1972)CrossRefzbMATHGoogle Scholar
  62. 62.
    Schneider, R.: Krümmungsschwerpunkte konvexer Körper. II. Abh. Math. Sem. Univ. Hambg. 37, 204–217 (1972)CrossRefzbMATHGoogle Scholar
  63. 63.
    Schneider, R.: Mixed polytopes. Discrete Comput. Geom. 29, 575–593 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Schneider, R.: Local tensor valuations on convex polytopes. Monatsh. Math. 171, 459–479 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  66. 66.
    Schneider, R., Schuster, R.: Tensor valuations on convex bodies and integral geometry. II. In: IV International Conference in “Stochastic Geometry, Convex Bodies, Empirical Measures and Applications to Engineering Science”, vol. II (Tropea, 2001). Rend. Circ. Mat. Palermo (2) Suppl. No. 70, part II, 295–314 (2002)Google Scholar
  67. 67.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  68. 68.
    Schröder-Turk, G.E., Kapfer, S., Breidenbach, B., Beisbart, C., Mecke, K.: Tensorial Minkowski functionals and anisotropy measures for planar patterns. J. Microsc. 238, 57–74 (2010)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Schröder-Turk, G.E., Mickel, W., Kapfer, S.C., Klatt, M.A., Schaller, F.M., Hoffmann, M.J.F., Kleppmann, N., Armstrong, P., Inayat, A., Hug, D., Reichelsdorfer, M., Peukert, W., Schwieger, W., Mecke, K.: Minkowski tensor shape analysis of cellular, granular and porous structures. Adv. Mater. Spec. Issue Hierarchical Struct. Towar. Funct. 23, 2535–2553 (2011)Google Scholar
  70. 70.
    Schütrumpf, B., Klatt, M.A., Iida, K., Schröder-Turk, G.E., Maruhn, J.A., Mecke, K., Reinhard, P.G.: Appearance of the single gyroid network phase in “nuclear pasta” matter. Phys. Rev. C 91, 025801 (2015)CrossRefGoogle Scholar
  71. 71.
    Schulte, J., Kousholt, A.: Reconstruction of convex bodies from moments. Discrete Comput. Geom. (to appear) (2016). arXiv:1605.06362
  72. 72.
    Schulte, J., Weil, W.: Valuations and Boolean models. In: Kiderlen, M., Vedel Jensen, E.B. (eds.) Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177. Springer, Berlin (2017)Google Scholar
  73. 73.
    Schuster, F.: Crofton measures and Minkowski valuations. Duke Math. J. 154, 1–30 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Schuster, F., Wannerer, T.: Minkowski valuations and generalized valuations. J. Eur. Math. Soc. (JEMS) (to appear) (2015). arXiv:1507.05412
  75. 75.
    Solanes, G.: Contact measures in isotropic spaces. Adv. Math. 317, 645–664 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Svane, A.M., Vedel Jensen, E.B.: Rotational Crofton formulae for Minkowski tensors and some affine counterparts. Adv. Appl. Math. 91, 44–75 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Vedel Jensen, E.B., Rasmusson, A.: Rotational integral geometry and local stereology -with a view to image analysis. In: Schmidt V (ed) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol. 2120, pp. 233–255. Springer, Cham (2015)Google Scholar
  78. 78.
    Wannerer, T.: The module of unitarily invariant area measures. J. Differ. Geom. 96, 141–182 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Wannerer, T.: Integral geometry of unitary area measures. Adv. Math. 263, 1–44 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Weil, W.: Translative and kinematic integral formulae for support functions. Geom. Dedicata 57, 91–103 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Weil, W.: Integral geometry of translation invariant functionals, I: the polytopal case. Adv. Appl. Math. 66, 46–79 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Weil, W.: Integral geometry of translation invariant functionals, II: the case of general convex bodies. Adv. Appl. Math. 83, 145–171 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Weis, J.A.: Tensorial curvature measures in integral geometry. Ph.D. Thesis, Karlsruhe Institute of Technology, Karlsruhe (2017). https://publikationen.bibliothek.kit.edu/1000071928/4245450
  84. 84.
    Wittmann, R., Marechal, M., Mecke, K.: Fundamental measure theory for smectic phases: scaling behavior and higher order terms. J. Chem. Phys. 141, 064103 (2014)CrossRefGoogle Scholar
  85. 85.
    Xia, C., Cao, Y., Kou, B., Li, J., Wang, Y., Xiao, X., Fezzaa, K.: Angularly anisotropic correlation in granular packings. Phys. Rev. E 90, 062201-1–062201-7 (2014)Google Scholar
  86. 86.
    Ziegel, J.F., Nyengaard, J.R., Vedel Jensen, E.B.: Estimating particle shape and orientation using volume tensors. Scand. J. Stat. 42, 813–831 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations