Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T00:50:31.890Z Has data issue: false hasContentIssue false
  • English
  • Français

On Multiple Integral Geometric Integrals and Their Applications to Probability Theory

Published online by Cambridge University Press:  20 November 2018

Franz Streit
Franz Streit*
Affiliation:
University of Toronto, Toronto, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It has been pointed out repeatedly in the literature that the methods of integral geometry (a mathematical theory founded by Wilhelm Blaschke and considerably extended by several mathematicians) provide highly suitable means for the solution of problems concerning “geometrical probabilities“ [2; 6; 12; 15]. The possibilities for the application of these integral geometric results to the evaluation of probabilities, satisfying certain conditions of invariance with respect to a group of transformations which acts on the probability space, are obviously not yet exhausted. In this article, such applications are presented. First, some concepts and notation are introduced (§1). In the next section we derive some integral geometric relations (§ 2). These results are generalizations of known systems of formulae and they are valid in the k-dimensional Euclidean space. In § 3, we determine mean-value formulae for the fundamental characteristics of point-sets, generated by randomly placed convex bodies.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 151 - 163
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Blaschke, W., Vorlesungen iiber Integralgeometrie. I und II. (Teubner, Leipzig, 1936-37; Chelsea, New York, 1949; VEB Deutscher Verlag der Wissenschaften, Berlin, 1955).Google Scholar
2. L. Fejes, Töth and Hadwiger, H., Mittlere Treffzahlen und geometrische Wahrscheinlichkeiten, Experientia 3 (1947), 366–369.Google Scholar
3. Griinbaum, B., Convex polytopes (Interscience, London, 1967).Google Scholar
4. Hadwiger, H., Ueber die mittlere mittlere Breite zufallsartig gestalteter Polygone, Comment. Math. Helv. 11 (1938), 321–329.Google Scholar
5. Hadwiger, H., Ueber Mittelwerte im Figurengitter, Comment. Math. Helv. 11 (1939), 221–233.Google Scholar
6. Hadwiger, H., Altes und Neues iiber konvexe Körper (Birkhâuser, Basel, 1955).Google Scholar
7. Hadwiger, H., Eulers Charakteristik und kombinatorische Géométrie. J. Reine Angew. Math. 194 (1955), 101–110.Google Scholar
8. Hadwiger, H., Vorlesungen iiber Inhalt, Oberflàche und Isoperimetrie (Springer, Berlin, 1957).Google Scholar
9. Hadwiger, H., Zur Axiomatik der innermathematischen Wahrscheinlichkeitstheorie, Mitt. Verein. Schweiz. Versich.-Math. 58 (1958), 159–165.Google Scholar
10. Hadwiger, H., Normale Körper im euklidischen Raum und ihre topologischen und metrischen Eigenschaften, Math. Z. 71 (1959), 124–140.Google Scholar
11. Hadwiger, H. and Streit, F., Ueber Wahrscheinlichkeiten raumlicher Biindelungserscheinungen, Monatsh. Math. 74 (1970), 30–40.Google Scholar
12. Kendall, M. G. and Moran, P. A. P., Geometrical probability (Griffin, London, 1963).Google Scholar
13. Santalö, L. A., Geometria Integral. 4: Sobre la medida cinemdtica en el piano, Abh. Math. Sem. Univ. Hamburg 11 (1936), 222–236.Google Scholar
14. Santalö, L. A., Integralgeometrie. 5: Ueber das kinematische Mass imRaum (Hermann, Paris, 1936).Google Scholar
15. Santalö, L. A., Introduction to integral geometry (Hermann, Paris, 1953).Google Scholar
16. Streit, F., Anordnungswahrscheinlichkeiten bei poissonverteilten Ereignissen, Ph.D. Thesis, Universitât Bern, Bern, Switzerland, 1966.Google Scholar