A local steiner-type formula for general closed sets and applications

Mathematische Zeitschrift

Volumn 246, Issue 1-2, 2004, Pages 237-272

  Hug, Daniel ^a   Last, Günter ^b   Weil, Wolfgang ^b  

^a UNIVERSITY OF FREIBURG   (Germany)

^b UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0742288806     PISSN: 00255874     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00209-003-0597-9     Document Type: Article

References (35)

1. Aleksandrov, A.D.: Zur Theorie der gemischten Volumina von konvexen Körpern, 1. Verallgemeinerung einiger Begriffe der Theorie der konvexen Körper (in Russian), Mat. Sbornik N.S. 2, 947-972 (1937)
2. Baddeley, A.J., Gill, R.D.: The empty space hazard of a spatial pattern. University Utrecht, Department of Math. Preprint 845, (1994)
3. Falconer, K.: Fractal Geometry, Mathematical Foundations and Applications. Wiley, Chichester, 1990
4. Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418-491 (1959)
5. Federer, H.: Geometric Measure Theory. Springer, Berlin, 1969
6. Fenchel, W., Jessen, B.: Mengenfunktionen und konvexe Körper. Danske Vid. Selsk., Mat.-Fys. Medd. 16, 1-31 (1938)
7. Ferry, S.: When ε-boundaries are manifolds. Fund. Math. 90, 199-210 (1975/76)
8. Fremlin, D.H.: Skeletons and central sets. Proc. London Math. Soc. 74, 701-720 (1997)
9. Fu, J.H.G.: Tubular neighborhoods in Euclidean spaces. Duke Math. J. 52, 1025-1046 (1985)
10. Gariepy, R., Pepe, W.D.: On the level sets of a distance function in a Minkowski space. Proc. Am. Math. Soc. 31, 255-259 (1972)
11. Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. I. Cartesian currents. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 37, Springer-Verlag, Berlin, 1998
12. Gray, A.: Tubes. Addison-Wesley, Redwood City, 1990
13. Hansen, M.B., Gill, R.D., Baddeley, A.J.: Kaplan-Meier type estimators for linear contact distributions. Scand. J. Statist. 23, 129-155 (1998)
14. Howard, R.: Boundary structure, comparison theorems, and strong maximum principles for sets with locally positive inner support radius. Preliminary Draft, October 2002
15. Hug, D.: Generalized curvature measures and singularities of sets with positive reach. Forum Math. 10, 699-728 (1998)
16. Hug, D.: Measures, curvatures and currents in convex geometry, Habilitationsschrift. Universität Freiburg, Freiburg, Dezember 1999
17. Hug, D., Last, G.: On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Probab. 28, 796-850 (2000)
18. Hug, D., Last, G., Weil, W.: Generalized contact distributions of inhomogeneous Boolean models. Adv. in Appl. Probab. 34, 21-47 (2002)
19. Hug, D., Last, G., Weil, W.: A survey on contact distributions. In: Morphology of Condensed Matter, K. Mecke, D. Stoyan, (eds.), Lecture Notes Phys, Springer Berlin, 600, 317-357 (2002)
20. Kallenberg, O.: Random Measures. Akademie-Verlag Berlin and Academic Press, London, 1983
21. Kneser, M.: Über den Rand von Parallelkörpern. Math. Nachr. 5, 241-251 (1951)
22. Last, G., Schassberger, R.: On the distribution of the spherical contact vector of stationary germ-grain models. Adv. Appl. Probab. 30, 36-52 (1998)
23. Matheron, G.: Random Sets and Integral Geometry. Wiley, New York, 1975
24. Rataj, J., Zähle, M.: Curvatures and currents for unions of sets with positive reach II. Ann. Global Anal. Geom. 20, 1-21 (2001)
25. Schneider, R.: Bestimmung konvexer Körper durch Krümmungsmaße. Comment. Math. Helv. 54, 42-60 (1979)
26. Schneider, R.: Parallelmengen mit Vielfachheit und Steiner-Formeln. Geom. Dedicata. 9, 111-127 (1980)
27. Schneider, R.: Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, 1993
28. Simon, L.M.: Lecture on Geometric Measure Theory. Australian National University, Canberra, 1984
29. Stachó, L.L.: On the volume function of parallel sets. Acta Sci. Math. 38, 365-374 (1976)
30. Stachó, L.L.: On curvature measures. Acta Sci. Math. 41, 191-207 (1979)
31. Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications. Second Edition, Wiley, Chichester, 1995
32. Walter, R.: Some analytical properties of geodesically convex sets. Abh. Math. Sem. Univ. Hamburg 45, 263-282 (1976)
33. Weil, W.: Mean bodies associated with random closed sets. Suppl. Rend. Circ. Mat. Palermo, Ser. II 50, 387-412 (1997)
34. Weyl, H.: On the volume of tubes. Am. J. Math. 61, 461-472 (1939)
35. Zähle, M.: Integral and current representation of Federer's curvature measures. Arch. Math. 46, 557-567 (1986)