On the variance of random polygons

Imre Bárány, William Steiger

Research output: Contribution to conferencePaperpeer-review

1 Scopus citations


A random polygon is the convex hull of uniformly dis-tributed random points in a convex body K ⊂ R2. Gen- eral upper bounds are established for the variance of the area of a random polygon and also for the variance of its number of vertices. The upper bounds have the same order of magnitude as the known lower bounds on vari-Ance for these functionals. The results imply a strong law of large numbers for the area and number of ver- Tices of random polygons for all planar convex bodies. Similar results had been known, but only in the special cases when K is a polygon or where K is a smooth con- vex body. The careful, technical arguments we needed may lead to tools for analogous extensions to general convex bodies in higher dimension. On the other hand one of the main results is a stronger version in dimension d = 2 of the economic cap covering theorem of Barany and Larman. It is crucial to our proof, but it does not extend to higher dimension.

Original languageEnglish (US)
Number of pages4
StatePublished - 2010
Event22nd Annual Canadian Conference on Computational Geometry, CCCG 2010 - Winnipeg, MB, Canada
Duration: Aug 9 2010Aug 11 2010


Other22nd Annual Canadian Conference on Computational Geometry, CCCG 2010
CityWinnipeg, MB

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Geometry and Topology


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