Advertisement

Acta Mathematica Hungarica

, Volume 155, Issue 1, pp 3–24 | Cite as

Intersection probabilities and kinematic formulas for polyhedral cones

  • R. Schneider
Article
  • 27 Downloads

Abstract

For polyhedral convex cones in \({\mathbb{R}^d}\), we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic random central hyperplane arrangement, we find probabilities for non-trivial intersection, either with a fixed cone, or for two independent random cones of this type.

Key words and phrases

polyhedral cone conic curvature measure kinematic formula intersection probability central hyperplane arrangement 

Mathematics Subject Classification

primary 52A22 secondary 52A55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Amelunxen, Measures on polyhedral cones: characterizations and kinematic formulas (preprint), arXiv:1412.1569v2 (2015)
  2. 2.
    D. Amelunxen and P. Bürgisser, Intrinsic volumes of symmetric cones, (extended version of [3]), arXiv:1205.1863 (2012)
  3. 3.
    D. Amelunxen and P. Bürgisser, Intrinsic volumes of symmetric cones and applications in convex programming, Math. Programm., Ser. A , 149 (2015), 105–130Google Scholar
  4. 4.
    Amelunxen, D., Lotz, M.: Intrinsic volumes of polyhedral cones: a combinatorial perspective. Discrete Comput. Geom. 58, 371–409 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amelunxen, D., Lotz, M., McCoy, M.B., Tropp, J.A.: Living on the edge: phase transitions in convex programs with random data. Inf. Inference 3, 224–294 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cover, T.M., Efron, B.: Geometrical probability and random points on a hypersphere. Ann. Math. Statist. 38, 213–220 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Glasauer, Integralgeometrie konvexer Körper im sphärischen Raum, Doctoral Thesis, Albert-Ludwigs-Universität (Freiburg i. Br, 1995); available from: www.hs-augsburg.de/~glasauer/publ/diss.pdf
  8. 8.
    Glasauer, S.: Integral geometry of spherically convex bodies. Diss. Summ. Math. 1, 219–226 (1996)MathSciNetGoogle Scholar
  9. 9.
    Hug, D., Schneider, R.: Random conical tessellations. Discrete Comput. Geom. 56, 395–426 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hug, D., Schneider, R., Schuster, R.: Integral geometry of tensor valuations. Adv. Appl. Math. 41, 482–509 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Hug and J. A. Weis, Kinematic formulae for tensorial curvature measures (preprint), arXiv:1612.08427 (2016)
  12. 12.
    McCoy, M.B., Tropp, J.A.: Sharp recovery bounds for convex demixing, with applications. Found. Comput. Math. 14, 503–567 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    McCoy, M.B., Tropp, J.A.: From Steiner formulas for cones to concentration of intrinsic volumes. Discrete Comput. Geom. 51, 926–963 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. B. McCoy and J. A. Tropp, The achievable performance of convex demixing (preprint), arXiv:1309.7478 (2013)
  15. 15.
    McMullen, P.: Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Camb. Philos. Soc. 78, 247–261 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Santaló, L.A.: Sobre la formula de Gauss-Bonnet para poliedros en espacios de curvatura constante. Revista Un. Mat. Argentina 20, 79–91 (1962)MathSciNetGoogle Scholar
  17. 17.
    Santaló, L.A.: Sobre la formula fundamental cinematica de la geometria integral en espacios de curvatura constante. Math. Notae 18, 79–94 (1962)MathSciNetGoogle Scholar
  18. 18.
    Santaló, L.A.: Integral Geometry and Geometric Probability, Encyclopedia of Mathematics and Its Applications, vol. 1. MA, Addison-Wesley (Reading (1976)Google Scholar
  19. 19.
    Schneider, R.: Curvature measures of convex bodies. Ann. Mat. Pura Appl. 116, 101–134 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R. Schneider, Convex Bodies: The Brunn–Minkowski Theory , 2nd ed., Encyclopedia of Mathematics and Its Applications, vol. 151, Cambridge University Press (Cambridge, 2014)Google Scholar
  21. 21.
    R. Schneider and W. Weil, Stochastic and Integral Geometry , Springer (Berlin, 2008)Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutAlbert-Ludwigs-UniversitätFreiburg i. Br.Germany

Personalised recommendations