Summary
For a wide class of stationary random hypersurfaces in ℝd the notion of the projection body is introduced. It turns out that this convex body, a very special case of which is Matheron's “Steiner compact” associated with a Poisson process of hyperplanes, contains most of the information concerning certain intersection properties of the random hypersurface, while its polar reciprocal set is closely connected with the behaviour of the random hypersurface in visibility problems. This enables one to give a unified treatment of several intersection and visibility problems for random hypersurfaces. A detailed investigation of the case where the random hypersurface is generated by a Poisson process is given separately.
References
Aleksandrov, A.D.: Zur Theorie der gemischten Volumina von konvexen Körpern I. Verallgemeinerung einiger Begriffe der Theorie der konvexen Körper. Mat. Sbornik44 N.S. 2, 947–972 (1937)
Blaschke, W.: Uber Schiebungen. Math. Z.42, 399–410 (1937)
Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. (Nachdruck) Berlin-Heidelberg-New York:Springer 1974
Busemann, H.: Convex surfaces. New York: Interscience Publ. 1958
Federer, H.: Some integralgeometric theorems. Trans. Am.Math. Soc.77, 238–261 (1954)
Federer, H.: Geometric measure theory. Berlin-Heidelberg-New York: Springer 1969
Federer, H.: Colloquium on geometric measure theory. Bull. Am. Math. Soc.84, 291–338 (1978)
Goodey, P. R., Woodcock, M.M.: Intersections of convex bodies with their translates. In: Davis, G., Grünbaum, B., Scherk, P. (eds.). The geometric vein. New York-Heidelberg-Berlin: Springer 1982
Hadwiger, H.: Altes und Neues über konvexe Körper. Basel-Stuttgart: Birkhäuser 1955
Leichtweiß, K.: Konvexe Mengen. Berlin-Heidelberg-New York. Springer 1980
Matheron, G.: Random sets and integral geometry. New York: Wiley 1975
Miles, R.E.: Random polytopes: the generalisation ton dimensions of the intervals of a Poisson process. Ph.D. Thesis, Cambridge University (1961)
Miles, R.E.: Poisson flats in euclidean spaces, Part I. Adv. Appl. Probab.1, 211–237 (1969)
Miles, R.E.: Poisson flats in euclidean spaces, Part II. Adv. Appl. Probab.3, 1–43 (1971)
Neveu, J.: Processus ponctuels. In: Lect. Notes Math.598. Berlin-Heidelberg-New York: Springer 1977
Petty, C.M.: Affine isoperimetric problems. In: Goodman, J.E., Lutwak, E., Malkevitch, J., Pollak, R. (eds.). Discrete geometry and convexity. New York Academy of Sciences. [To be published]
Pólya, G.: Zahlentheoretisches und Wahrscheinlichkeitstheoretisches über die Sichtweite im Walde. Arch. Math. Phys.27, 135–142 (1918)
Saint Raymond, J.: Sur le volume des corps convexes symétriques. Publ. Math. Univ. Pierre Marie Curie46 (1981)
Santaló, L.A.: Integral geometry and geometric probability. Encyclopedia of Mathematics and its applications I. Reading, Mass.: Addison-Wesley 1976
Santaló, L.A.: Sobre la distribucion probable de corpusculos en un cuerpo deducida de la distribucion en sus secciones y problemas analogos. Rev. Union Mat. Argent9, 145–164 (1943)
Schneider, R.: Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitstheor. Verw. Geb.61, 379–387 (1982)
Schneider, R.: Random polytopes generated by anisotropic hyperplanes. Bull. Lond. Math. Soc.14, 549–553 (1982)
Serra, J.: Image analysis and mathematical morphology. London: Academic press 1982
Thomas, C.: Extremum properties of the intersection densities of stationary Poisson hyperplane processes. Math. Operationsforsch. u. Statist. (ser. statist.)15, 443–449 (1984)
Weil, W., Wieacker, J.A.: Densities for stationary random sets and point processes. Adv. Appl. Probab.16, 324–346 (1984)
Wieacker, J.A.: Translative Poincaré formulae for Hausdorff rectifiable sets. Geom. Dedicata16, 231–248 (1984)
Zähle, M.: Random processes of Hausdorff rectifiable closed sets. Math. Nachr.108, 49–72 (1982)
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Wieacker, J.A. Intersections of random hypersurfaces and visibility. Probab. Th. Rel. Fields 71, 405–433 (1986). https://doi.org/10.1007/BF01000214
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DOI: https://doi.org/10.1007/BF01000214
Keywords
- Stochastic Process
- Probability Theory
- Poisson Process
- Mathematical Biology
- Convex Body