Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-20T01:03:12.208Z Has data issue: false hasContentIssue false
  • English
  • Français

On the convex hull of random points in a polytope

Published online by Cambridge University Press:  14 July 2016

Rex A. Dwyer
Rex A. Dwyer*
Affiliation:
Carnegie-Mellon University
*
∗Present address: Computer Science Department, North Carolina State University, Raleigh, NC 27695–8206, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(logd–1n) for any polytope, the expected number of vertices is Ω(logd–1n) for any simple polytope, and the expected number of facets is O(logd–1n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.

Keywords

EXTREME POINTSGEOMETRIC PROBABILITYAVERAGE-CASE ANALYSIS OF ALGORITHMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 4 , December 1988 , pp. 688 - 699
Copyright
Copyright © Applied Probability Trust 1988 

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

Supported by National Science Foundation Grant No. ECS-8418392.

This paper is dedicated to the memory of Professor Rebecca S. Nelson, esteemed teacher.

References

[1] Bentley, J. L. and Shamos, M. I. (1978) Divide and conquer for linear expected time. Informat. Proc. Letters 7, 87–91.Google Scholar
[2] Bentley, J. L., Kung, H. T., Schkolnick, M. and Thompson, C. D. (1978) On the average number of maxima in a set of vectors. J. Assoc. Comput. Mach. 25, 536–543.Google Scholar
[3] Brønsted, A. (1983) An Introduction to Convex Polytopes. Springer-Verlag, New York.CrossRefGoogle Scholar
[4] Buchta, C. (1985) Zufallige Polyeder: eine Übersicht. In Zahlentheoretische Analysis , ed. Hlawka, E., Springer-Verlag, Berlin, 1–13.Google Scholar
[5] Buchta, C, Müller, F. and Tichy, R. F. (1985) Stochastical approximation of convex bodies. Math. Annalen 271, 225–235.Google Scholar
[6] Carnal, H. (1970) Die konvexe Hülle von n rotationssymmetrisch verteilten Punkten. Z. Wahrscheinlichkeitsth. 15, 168–176.CrossRefGoogle Scholar
[7] Devroye, L. P. (1980) A note on finding convex hulls via maximal vectors. Informat. Proc. Letters 11, 53–56.CrossRefGoogle Scholar
[8] Devroye, L. P. (1981) How to reduce the average complexity of convex hull finding algorithms. Comput. Math. Appl. 7, 299–308.Google Scholar
[9] Devroye, L. P. (1983) Moment inequalities for random variables in computational geometry. Computing 30, 111–119.CrossRefGoogle Scholar
[10] Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331–342.Google Scholar
[11] Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. 2, 2nd edn. Wiley, New York.Google Scholar
[12] Megiddo, N. (1984) Linear programming in linear time when the dimension is fixed. J. Assoc. Comput. Mach. 31, 114–127.CrossRefGoogle Scholar
[13] Preparata, F. P. and Shamos, M. I. (1985) Computational Geometry: An Introduction. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[14] Raynaud, H. (1970) Sur l'enveloppe convexe des nuages des points aléatoires dans Rn. I. J. Appl. Prob. 7, 35–48.Google Scholar
[15] Renyi, A. and Sulanke, R. (1963) Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 2, 75–84.CrossRefGoogle Scholar
[16] Renyi, A. and Sulanke, R. (1964) Über die konvexe Hülle von n zufallig gewählten Punkten, II. Z. Wahrscheinlichkeitsth. 3, 138–147 Google Scholar
[17] Seidel, R. (1986) Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proc. 18th ACM Symp. Theory of Computing , 404–413.CrossRefGoogle Scholar