scholarly journals Exact Formulae for Variances of Functionals of Convex Hulls

2013 ◽  
Vol 45 (04) ◽  
pp. 917-924
Author(s):  
Christian Buchta
Keyword(s):  
Convex Hull ◽  
Asymptotic Distribution ◽  
Convex Polygon ◽  
Convex Hulls ◽  
Affine Invariance ◽  
Uniform Sample

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).

scholarly journals Exact Formulae for Variances of Functionals of Convex Hulls

2013 ◽  
Vol 45 (4) ◽  
pp. 917-924 ◽  
Author(s):  
Christian Buchta
Keyword(s):  
Convex Hull ◽  
Asymptotic Distribution ◽  
Convex Polygon ◽  
Convex Hulls ◽  
Affine Invariance ◽  
Uniform Sample

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).

scholarly journals Joint Distribution of the Number of Vertices and the Area of Convex Hulls Generated by a Uniform Distribution in a Convex Polygon

2021 ◽  
pp. 232-243
Author(s):  
Isakjan M. Khamdamov ◽  
Zoya S. Chay
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convex Hull ◽  
Uniform Distribution ◽  
Joint Distribution ◽  
Poisson Point Process ◽  
Convex Polygon ◽  
Point Processes ◽  
Poisson Approximation ◽  
Convex Hulls

A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process

Random convex hulls: a variance revisited

2004 ◽  
Vol 36 (04) ◽  
pp. 981-986
Author(s):  
Steven Finch ◽  
Irene Hueter
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convex Hull ◽  
Asymptotic Variance ◽  
Circular Disk ◽  
Exact Expression ◽  
Convex Hulls ◽  
Asymptotic Constant ◽  
Uniform Sample ◽  
Random Convex

An exact expression is determined for the asymptotic constant c 2 in the limit theorem by P. Groeneboom (1988), which states that the number of vertices of the convex hull of a uniform sample of n random points from a circular disk satisfies a central limit theorem, as n → ∞, with asymptotic variance 2πc 2 n 1/3.

scholarly journals Convex Hulls of Uniform Samples from a Convex Polygon

2012 ◽  
Vol 44 (2) ◽  
pp. 330-342 ◽  
Author(s):  
Piet Groeneboom
Keyword(s):  
Convex Hull ◽  
Central Limit ◽  
Convex Polygon ◽  
Poisson Approximation ◽  
Standard Normal Distribution ◽  
Asymptotic Results ◽  
Uniform Sample ◽  
Exponential Random Variables ◽  
Sample Variances ◽  
Limit Result

In Groeneboom (1988) a central limit theorem for the number of vertices Nn of the convex hull of a uniform sample from the interior of a convex polygon was derived. To be more precise, it was shown that {Nn - (2/3)rlogn} / {(10/27)rlogn}1/2 converges in law to a standard normal distribution, if r is the number of vertices of the convex polygon from which the sample is taken. In the unpublished preprint Nagaev and Khamdamov (1991) a central limit result for the joint distribution of Nn and An is given, where An is the area of the convex hull, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponential random variables. We derive this representation from the representation in Groeneboom (1988) and also prove the central limit result of Nagaev and Khamdamov (1991), using this representation. The relation between the variances of the asymptotic normal distributions of the number of vertices and the area, established in Nagaev and Khamdamov (1991), corresponds to a relation between the actual sample variances of Nn and An in Buchta (2005). We show how these asymptotic results all follow from one simple guiding principle. This corrects at the same time the scaling constants in Cabo and Groeneboom (1994) and Nagaev (1995).

scholarly journals Convex Hulls of Uniform Samples from a Convex Polygon

2012 ◽  
Vol 44 (02) ◽  
pp. 330-342 ◽  
Author(s):  
Piet Groeneboom
Keyword(s):  
Convex Hull ◽  
Central Limit ◽  
Convex Polygon ◽  
Poisson Approximation ◽  
Standard Normal Distribution ◽  
Asymptotic Results ◽  
Uniform Sample ◽  
Exponential Random Variables ◽  
Sample Variances ◽  
Limit Result

In Groeneboom (1988) a central limit theorem for the number of vertices N n of the convex hull of a uniform sample from the interior of a convex polygon was derived. To be more precise, it was shown that {N n - (2/3)rlogn} / {(10/27)rlogn}1/2 converges in law to a standard normal distribution, if r is the number of vertices of the convex polygon from which the sample is taken. In the unpublished preprint Nagaev and Khamdamov (1991) a central limit result for the joint distribution of N n and A n is given, where A n is the area of the convex hull, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponential random variables. We derive this representation from the representation in Groeneboom (1988) and also prove the central limit result of Nagaev and Khamdamov (1991), using this representation. The relation between the variances of the asymptotic normal distributions of the number of vertices and the area, established in Nagaev and Khamdamov (1991), corresponds to a relation between the actual sample variances of N n and A n in Buchta (2005). We show how these asymptotic results all follow from one simple guiding principle. This corrects at the same time the scaling constants in Cabo and Groeneboom (1994) and Nagaev (1995).

Random convex hulls: a variance revisited

2004 ◽  
Vol 36 (4) ◽  
pp. 981-986 ◽  
Author(s):  
Steven Finch ◽  
Irene Hueter
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convex Hull ◽  
Asymptotic Variance ◽  
Circular Disk ◽  
Exact Expression ◽  
Convex Hulls ◽  
Asymptotic Constant ◽  
Uniform Sample ◽  
Random Convex

An exact expression is determined for the asymptotic constant c2 in the limit theorem by P. Groeneboom (1988), which states that the number of vertices of the convex hull of a uniform sample of n random points from a circular disk satisfies a central limit theorem, as n → ∞, with asymptotic variance 2πc2n1/3.

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

scholarly journals COMPUTING CONVEX HULLS BY AUTOMATA ITERATION

2009 ◽  
Vol 20 (04) ◽  
pp. 647-667
Author(s):  
FRANÇOIS CANTIN ◽  
AXEL LEGAY ◽  
PIERRE WOLPER
Keyword(s):  
Convex Hull ◽  
Finite Automaton ◽  
Convex Hulls ◽  
The Real ◽  
Starting Point ◽  
Finite Set

This paper considers the problem of computing the real convex hull of a finite set of n-dimensional integer vectors. The starting point is a finite-automaton representation of the initial set of vectors. The proposed method consists in computing a sequence of automata representing approximations of the convex hull and using extrapolation techniques to compute the limit of this sequence. The convex hull can then be directly computed from this limit in the form of an automaton-based representation of the corresponding set of real vectors. The technique is quite general and has been implemented.

The convex hull of a uniform sample from the interior of a simple d-polytope

1989 ◽  
Vol 26 (02) ◽  
pp. 259-273 ◽  
Author(s):  
Barthold F. Van Wel
Keyword(s):  
Convex Hull ◽  
Asymptotic Expression ◽  
Expected Number ◽  
Simple Polytope ◽  
Uniform Sample ◽  
Sample Points

An asymptotic expression is given for the expected number of vertices of the convex hull of a uniform sample from the interior of a d-dimensional simple polytope. This extends a result derived by Rényi and Sulanke for sample points in the plane.

CONVEX HULLS OF SIERPIŃSKI RELATIVES

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850098
Author(s):  
T. D. TAYLOR ◽  
S. ROWLEY
Keyword(s):  
Fractal Dimension ◽  
Convex Hull ◽  
Convex Hulls ◽  
Symmetry Properties

This paper presents an investigation of the convex hulls of the Sierpiński relatives. These fractals all have the same fractal dimension but different topologies. We prove that the relatives have convex hulls with polygonal boundaries with at most 12 vertices. We provide a method for finding the convex hull of a relative using its scaling and symmetry properties and present examples. We also investigate the connectivity properties of certain classes of relatives with the same convex hulls.

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