Skip to main content
Log in

Geometric Inequalities for Poisson Processes of Convex Bodies and Cylinders

  • Published:
Results in Mathematics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. H. Busemann, Convex surfaces. Interscience Publishers. New York 1958.

    MATH  Google Scholar 

  2. P. Davy, Projected thick sections through multi-dimensional particle aggregates. J. Appl. Prob. 13 (1976), 714–722. Correction: J. Appl. Prob. 15 (1978), 456.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Davy, Stereology - A statistical viewpoint. Thesis, Canberra 1978.

  4. W.J. Firey, Blaschke sums of convex bodies and mixed bodies. Proc. Colloquium Convexity (Copenhagen 1965), Københavns Univ. Mat. Inst. 1967, 94–101.

  5. W.J. Firey, Support flats to convex bodies. Geom. Dedicata 2 (1973), 225–248.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Giesler, Ungleichungen für die Volumina von Polar- und Projektionenkörpern. Diplomarbeit, Freiburg 1984.

  7. H.J.G. Guhdersen, Estimation of tubule or cylinder LV,SV and VV on thick sections. J. Microscopy 117 (1979), 333–345.

    Article  Google Scholar 

  8. E. Heil, Ungleichungen für die Quermaßintegrale polarer Körper. Manuscripta Math. 19 (1976), 143–149.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Janson, O. Kallenberg, Maximizing the intersection density of fibre processes. J. Appl. Prob. 18 (1981), 820–828.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Kneser, W. Süss, Die Volumina in linearen Scharen konvexer Körper. Mat. Tidsskr. B 1932, 19–25.

    Google Scholar 

  11. S.S. Kutateladze, Blaschke structures in the programming of isoperimetric structures (Russian) Mat. Zametki 14 (1973), 745–754. English translation: Math. Notes 14 (1973), 985–989 (1974).

    MathSciNet  MATH  Google Scholar 

  12. K. Leichtweiss, Konvexe Mengen. VEB Deutsch. Verl. d. Wiss., Berlin 1980; Springer-Verlag, Berlin etc. 1980.

    Book  Google Scholar 

  13. E. Lutwak, Dual mixed volumes. Pacific J. Math. 58 (1975), 531–538.

    Article  MathSciNet  MATH  Google Scholar 

  14. E. Lutwak, Cross-sectional measures of polar reciprocal convex bodies. Geom. Dedicata 5 (1976), 79–80.

    Article  MathSciNet  MATH  Google Scholar 

  15. E. Lutwak, Mixed projection inequalities. Trans. Amer. Math. Soc. 287 (1985), 91–106.

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Lutwak, On some affine isoperimetric inequalities. J. Differential Geom. (to appear)

  17. G. Matheron, Random sets and integral geometry. Wiley, New York etc. 1975.

    MATH  Google Scholar 

  18. G. Matheron, La formule de Crofton pour les sections épaisses. J. Appl. Prob. 13 1976, 707–713.

    Article  MathSciNet  MATH  Google Scholar 

  19. R.E. Miles, Poisson flats in Euclidean spaces, Part II: Homogeneous Poisson flats and the complementary theorem. Adv. Appl. Prob. 2 (1971), 1–43.

    Article  MathSciNet  Google Scholar 

  20. CM. Petty, Isoperimetric problems. Proc. Conf. Convexity and Combinat. Geom., Univ. of Oklahoma 1971 (1972), 26–41.

  21. S. Reisner, Random polytopes and the volume product of symmetric convex bodies. Math. Scand. (to appear)

  22. S. Reisner, Zonoids with minimal volume-product. (to appear)

  23. B.D. Ripley, The foundations of stochastic geometry. Ann. Prob. 4 (1976), 995–998.

    Article  MathSciNet  MATH  Google Scholar 

  24. L.A. Santaló, Probabilidades sobre cuerpos convexos y cilindros. Revista Un. Mat. Argentina 25 (1970), 95–104.

    MATH  Google Scholar 

  25. L.A. santaló, Integral geometry and geometric probability. Encyclopedia Math. Appl. vol.1, Addison–Wesley Publ. Co., Reading, Mass. 1976.

    Google Scholar 

  26. R. Schneider, Curvature measures of convex bodies. Ann. Mat. Pura Appl. JL16 (1978), 101–134.

    Article  Google Scholar 

  27. R. Schneider, Crofton’s formula generalized to projected thick sections. Rend. Circ. Mat. Palermo 30 (1981), 157–160.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. Schneider, Curvature measures and integral geometry of convex bodies. Rend. Sem. Mat. Univers. Politecn. Torino 38 (1980), 79–98.

    MATH  Google Scholar 

  29. R. Schneider, Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitsth. verw. Geb. 62. (1982), 379–387.

    Article  Google Scholar 

  30. R. Schneider, Random polytopes generated by anisotropic hyperplanes. Bull. London Math. Soc. 14 (1982), 549–553.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Schneider, W. Weil, Translative and kinematic integral formulae for curvature measures. Math. Nachr. (to appear)

  32. D. Stoyan, J. Mecke, Stochastische Geometrie. Akademie-Verlag, Berlin 1983.

    MATH  Google Scholar 

  33. C. Thomas, Extremum properties of the intersection densities of stationary Poisson hyperplane processes. Math. Operations forsch. Statist., ser. statist., 15 (1984), 443–449.

    MATH  Google Scholar 

  34. W. Weil, Über die Projektionenkörper konvexer Polytope. Arch. Math. 22 (1971), 664–672.

    Article  MathSciNet  MATH  Google Scholar 

  35. J.A. Wieacker, Translative Poincaré formulae for Hausdorff rectifiable sets. Geom. Dedicata 16 (1984), 231–248.

    Article  MathSciNet  MATH  Google Scholar 

  36. J.A. Wieacker, Intersections of random hypersurfaces and visibility. Z. Wahrscheinlichkeitsth. verw. Geb. (to appear)

  37. M. Zähle, Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108 (1982), 49–78.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schneider, R. Geometric Inequalities for Poisson Processes of Convex Bodies and Cylinders. Results. Math. 11, 165–185 (1987). https://doi.org/10.1007/BF03323266

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03323266

Keywords

Navigation