Exact Formulae for Variances of Functionals of Convex Hulls

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...

Full description

Saved in:
Bibliographic Details
Published in:Advances in applied probability 2013-12, Vol.45 (4), p.917-924
Main Author: Buchta, Christian
Format: Article
Language:English
Subjects:
52A22
60D05
Aircraft hulls
Asymptotic methods
Asymptotic properties
Convex analysis
Convex hull
Expected values
Functionals
Gaussian distributions
Graphs
Hulls
Hulls (structures)
Mathematical moments
Mathematical theorems
Origins
Pardons
Polygons
Polytopes
Probability
Probability distribution
random point
Stochastic Geometry and Statistical Applications
Studies
Triangles
variance
Variance analysis
Vertices
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 P 1,…, P n 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 Ñ n of points among P 1,…, P n , which are vertices of the convex hull of (0, 1), P 1,…, P n , and (1, 0). Correspondingly, D̃ n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var Ñ n 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).
ISSN:0001-8678
1475-6064
DOI:10.1239/aap/1386857850