Summary
The inclusion functional of a random convex set, evaluated at a fixed convex set K, measures the probability that the random convex set contains K. This functional is an analogue of the complement of the distribution function of an ordinary random variable. A methodology is described for evaluating the inclusion functional for the case where the random convex set is generated as the convex hull of n i.i.d. points from a distribution function F in the plane. For general K and F, the inclusion probability is difficult to compute in closed form. The case where K is a straight line segment is examined in detail and, in this situation, a simple answer is given for an interesting class of distributions F.
Article PDF
Similar content being viewed by others
References
Carnal, H.: Die konvexe Hülle von n rotationssymmetrisch verteilten Punkten. Z. Wahrscheinlichkeitstheorie verw. Gebiete 15, 168–179 (1970)
Crofton, M.W.: Probability. Encyclopaedia Brittanica, 9th ed., Vol. 19, 768–788 (1885)
Eddy, W.F.: The distribution of the convex hull of a Gaussian sample. J. Appl. Probability 17, 686–695 (1980)
Eddy, W.F., Trader, D.A.: Probability functionals for random sets. Technical Report No. 252, Department of Statistics, Carnegie-Mellon University, Pittsburgh (1982)
Efron, B.: The convex hull of a random set of points. Biometrika 52, 331–342 (1965)
Green, P.J.: Peeling bivariate data, in Interpreting Multivariate Data, ed., V. Barnett, New York; Wiley, pp. 3–19, 1981
Jewell, N.P., Romano, J.P.: Coverage problems and random convex hulls. J. Appl. Probability 19, 546–561 (1982)
Jewell, N.P., Romano, J.P.: The probability that a random convex hull contains a fixed set. Technical report, Program in Biostatistics, University of California, Berkeley (1982)
Kendall, D.G.: Foundation of a theory of random sets, in Stochastic Geometry, eds., E.F. Harding eds., E.F. Harding and D.G. Kendall, New York: Wiley, pp. 322–376 (1974)
Matheron, G.: Random Sets and Integral Geometry. New York: Wiley, 1975
Renyi, A., Sulanke, R.: Über die Konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie verw. Gebiete 2, 75–84 (1963)
Rogers, L.C.G.: The probability that two samples in the plane will have disjoint convex hulls. J. Appl. Probability 15, 790–802 (1978)
Solomon, H.: Geometric Probability. SIAM, Philadelphia (1978)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jewell, N.P., Romano, J.P. Evaluating inclusion functionals for random convex hulls. Z. Wahrscheinlichkeitstheorie verw Gebiete 68, 415–424 (1985). https://doi.org/10.1007/BF00535336
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00535336