Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T13:06:05.628Z Has data issue: false hasContentIssue false
  • English
  • Français

The average distance between two convex sets

Published online by Cambridge University Press:  14 July 2016

H. Groemer
H. Groemer*
Affiliation:
The University of Arizona
*
∗Postal address: Department of Mathematics, The University of Arizona, Tucson, AZ 85721, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In recent publications of Hadwiger, Bokowski, and Wills it has been shown that it is possible to evaluate the average distance between a fixed and a randomly distributed convex set in terms of the mean projection measures of these sets. In the present note it is pointed out that these and more general results can be obtained from a theorem concerning the relationship between integrals over arbitrary measure spaces and corresponding Lebesgue–Stieltjes integrals.

Keywords

AVERAGE DISTANCEINTEGRAL GEOMETRIC MEAN VALUESKINEMATIC PRINCIPAL FORMULAMIXED VOLUMES
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 415 - 422
Copyright
Copyright © Applied Probability Trust 1980 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by National Science Foundation Grant MCS 76–06111.

References

[1] Alemany, R. E. (1963) Valores numericos de ciertas constantes relacionadas con volumenes mixtos de cuerpos convexos. Rev. Un. Mat. Argentina 21, 113–118.Google Scholar
[2] Bokowski, J., Hadwiger, H. and Wills, J. M. (1976) Eine Erweiterung der Croftonschen Formeln für konvexe Körper. Mathematika 23, 212–219.Google Scholar
[3] Bonnesen, T. and Fenchel, W. (1934) Theorie der konvexen Körper. Springer-Verlag, Berlin.Google Scholar
[4] Groemer, H. (1977) On translative integral geometry. Arch. Math. (Basel) 29, 324–330.CrossRefGoogle Scholar
[5] Hadwiger, H. (1957) Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin.Google Scholar
[6] Hadwiger, H. (1975) Eine Erweiterung der kinematischen Hauptformel der Integralgeometrie. Abh. Math. Sem. Univ. Hamburg 44, 84–90.Google Scholar
[7] Hadwiger, H. (1975) Eikörperrichtungsfunktionale und kinematische Integralformeln. Lecture Notes, Bern.Google Scholar
[8] Loève, M. (1960) Probability Theory. Van Nos. and, Princeton, N.J.Google Scholar
[9] Nachbin, L. (1965) The Haar Integral. Van Nostrand, Princeton, N.J.Google Scholar
[10] Santaló, L. A. (1975) The kinematic formula in integral geometry for cylinders. Ann. Mat. Pura Appl. 103, 71–79.Google Scholar
[11] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
[12] Schneider, R. (1977) Eine kinematische Integralformel für konvexe Körper. Arch. Math. (Basel) 28, 217–220.Google Scholar
[13] Weil, W. (1979) Kinematic integral formulas for convex bodies. In Contributions to Geometry. Birkhäuser-Verlag, Basel.Google Scholar