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Exact Formulae for Variances of Functionals of Convex Hulls

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  04 January 2016

Christian Buchta
Christian Buchta*
Affiliation:
Salzburg University
*
∗ Postal address: Department of Mathematics, Salzburg University, Hellbrunner Straße 34, 5020 Salzburg, Austria. Email address: christian.buchta@sbg.ac.at
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Abstract

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The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider—in view of affine invariance—n points P1,…, Pn distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number Ñn of points among P1,…, Pn, which are vertices of the convex hull of (0, 1), P1,…, Pn, and (1, 0). Correspondingly, D̃n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var Ñn and var . These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev and Khamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).

Keywords

Convex hullrandom pointvariance

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 917 - 924
Copyright
© Applied Probability Trust 

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