The Annals of Probability

On support measures in Minkowski spaces and contact distributions in stochastic geometry

Daniel Hug and Günter Last

Full-text: Open access

Abstract

This paper is concerned with contact distribution functions of a random closed set $\Xi=\Bigcup_{n=1}^\infty \Xi_n$ in $\mathbb{R}^d$, where the $\Xi_n$ are assumed to be random nonempty convex bodies. These distribution functions are defined here in terms of a distance function which is associated with a strictly convex gauge body (structuring element) that contains the origin in its interior. Support measures with respect to such distances will be introduced and extended to sets in the local convex ring.These measures will then be used in a systematic way to derive and describe some of the basic properties of contact distribution functions. Most of the results are obtained in a general nonstationary setting.Only the final section deals with the stationary case.

Article information

Source
Ann. Probab. Volume 28, Number 2 (2000), 796-850.

Dates
First available: 18 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1019160261

Mathematical Reviews number (MathSciNet)
MR1782274

Digital Object Identifier
doi:10.1214/aop/1019160261

Zentralblatt MATH identifier
01906374

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G57: Random measures 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 46B20: Geometry and structure of normed linear spaces

Keywords
Stochastic geometry Minkowski space contact distribution function germ-grain model support (curvature) measure marked point process Palm probabilities randommeasure

Citation

Hug, Daniel; Last, Günter. On support measures in Minkowski spaces and contact distributions in stochastic geometry. The Annals of Probability 28 (2000), no. 2, 796--850. doi:10.1214/aop/1019160261. http://projecteuclid.org/euclid.aop/1019160261.


Export citation

References

  • [1] Baddeley, A. J. (1999). Spatial sampling and censoring. In Stochastic Geometry: Likelihood and Computation (O. Barndorff-Nielsen, W. Kendall and M. N. M. van Lieshout, eds.) 37-78. CRC Press, Boca Raton, FL.
  • [2] Baddeley, A. J. and Averback, P. (1983). Stereology of tubular structures. J. Microscopy 131 323-340.
  • [3] Baddeley, A. J. and Gill, R. D. (1994). The empty space hazard of a spatial pattern. Preprint 845, Dept. Mathematics, Univ. Utrecht.
  • [4] Chiu, S. N. and Stoyan, D. (1998). Estimators of distance distributions for spatial patterns. Statist. Neerlandica 52 239-246.
  • [5] Fallert, H. (1996). Quermaßdichten f ¨ur Punktprozesse konvexer K¨orper and Boolesche Modelle. Math. Nachr. 181 165-184.
  • [6] Fremlin, D. H. (1997). Skeletons and central sets. Proc. London Math. Soc. (3) 74 701-720.
  • [7] Groemer, H. (1978). On the extension of additive functionals on classes of convex sets. Pacific J. Math. 75 397-410.
  • [8] Hahn, U., Micheletti, A., Pohlink, R., Stoyan, D. and Wendrock, H. (1999). Stereological analysis and modelling of gradient structures. J. Microscopy 195 113-124.
  • [9] Hahn, U. and Stoyan, D. (1998). Unbiased stereological estimation of surface area density of gradient surface processes. Adv. in Appl. Probab. 30 904-920.
  • [10] Hansen, M. B., Baddeley, A. J. and Gill, R. D. (1999). First contact distributions for spatial patterns: regularity and estimation. Adv. in Appl. Probab. 31 15-33.
  • [11] Hansen, M. B., Gill, R. D. and Baddeley, A. J. (1998). Kaplan-Meier type estimators for linear contact distributions. Scand. J. Statist. 23 129-155.
  • [12] Heinrich, L. (1992). On existence and mixing properties of germ-grain models. Statistics 23 271-286.
  • [13] Heinrich, L. and Molchanov, I. S. (1999). Central limit theorems for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. 31 283-314.
  • [14] Hug, D. (2000). Contact distributions of Boolean models. Rend. Circ. Mat. Palerno 2.
  • [15] Kallenberg, O. (1993). Random Measures. Academic Press, London.
  • [16] Kiderlen, M. (1999). Schnittmittelungen und ¨aquivariante Endomorphismen konvexer K¨orper. Ph.D. dissertation, Univ. Karlsruhe.
  • [17] Kiderlen, M. and Weil, W. (1999). Measure-valued valuations and mixed curvature measures of convex bodies. Geom. Dedicata 76 291-329.
  • [18] Kingman, J. F. C. (1993). Poisson Processes. Oxford Univ. Press.
  • [19] Last, G. and Holtmann, M. (1999). On the empty space function of some germ-grain models. Pattern Recognition 32 1587-1600.
  • [20] Last, G. and Schassberger, R. (1998). On the distribution of the spherical contact vector of stationary germ-grain models. Adv. in Appl. Probab. 30 36-52.
  • [21] Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
  • [22] Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Wiley, Chichester.
  • [23] Mecke, J. (1967). Station¨are zuf¨allige Maße auf lokalkompakten Abelschen Gruppen.Wahrsch. Verw. Gebiete 9 36-58.
  • [24] Neveu, J. (1977). Processus ponctuels. Ecole d'Et´e de Probabilit´es de Saint-Flour VI. Lecture Notes in Math. 598 249-445. Springer, Berlin.
  • [25] Quintanella, J. and Torquato, S. (1997). Microstructure functions for a model of statistically inhomogeneous random media. Phys. Rev. E 55 1558-1565.
  • [26] Schneider, R. (1979). Bestimmung konvexer K¨orper durch Kr ¨ummungsmaße. Comment. Math. Helv. 54 42-60.
  • [27] Schneider, R. (1980). Parallelmengen mit Vielfachheit und Steiner-Formeln. Geom. Dedicata 9 111-127.
  • [28] Schneider, R. (1993). Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press.
  • [29] Schneider, R. (1994). An extension of the principal kinematic formula of integral geometry. Rend. Circ. Mat. Palermo (2) 35 275-290.
  • [30] Schneider, R. and Weil, W. (1992). Integralgeometrie. Teubner, Stuttgart.
  • [31] Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.
  • [32] Schneider, R. and Wieacker, J. A. (1997). Integral geometry in Minkowski spaces. Adv. Math. 129 222-260.
  • [33] Serra, J. (1988). Image Analysis and Mathematical Morphology 2: Theoretical Advances. Academic Press, London.
  • [34] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, Chichester.
  • [35] Thompson, A. C. (1996). Minkowski Geometry. Encyclopedia of Mathematics and Its Applications 63. Cambridge Univ. Press.
  • [36] Weil, W. (1981). Zuf¨allige Ber ¨uhrung konvexer K¨orper durch q-dimensionale Ebenen. Results Math. 4 84-101.
  • [37] Weil, W. (1997). The mean normal distribution of stationary random sets and particle processes. In Advances in Theory and Applications of Random Sets, (D. Jeulin, ed.) 21-33. World Scientific, Singapore.
  • [38] Weil, W. (1997). Mean bodies associated with randomclosed sets. Rend. Circ. Mat. Palermo (2) Suppl. 50 387-412.
  • [39] Weil, W. and Wieacker, J. A. (1984). Densities for stationary randomsets and point processes. Adv. in Appl. Probab. 16 324-346.
  • [40] Weil, W. and Wieacker, J. A. (1988). A representation theoremfor randomsets. Probab. Math. Statist. 9 147-151.
  • [41] Weil, W. and Wieacker, J. A. (1993). Stochastic geometry. In Handbook of Convex Geometry (P. M. Gruber and J. M. Wills, eds.) 1391-1438. North-Holland, Amsterdam.
  • [42] Z¨ahle, M. (1986). Curvature measures and random sets. II. Probab. Theory Related Fields 71 37-58.
  • [43] Z¨ahle, M. (1988). Randomcell complexes and generalized sets. Ann. Probab. 16 1742-1766.