The probability that two samples in the plane will have disjoint convex hulls

1978 ◽  
Vol 15 (4) ◽  
pp. 790-802 ◽  
Author(s):  
L. C. G. Rogers
Keyword(s):  
Convex Hull ◽  
Continuous Distribution ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Absolutely Continuous ◽  
Two Samples ◽  
Disjoint Convex ◽  
The Given

Suppose given an absolutely continuous distribution on the plane, and points Pi, · · ·, Pj, Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj the convex hull of {Pi, · · ·, Pj}, and by Γk the convex hull of (Πt, · · ·, Πk}, and writing pjk for the probability that Gj and Γk are disjoint, certain properties of the array {pjk; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p1n; n = 1,2, · · ·}, and asymptotic results for {pnn; n = 1,2, · · ·}. Some examples are considered.

The probability that two samples in the plane will have disjoint convex hulls

1978 ◽  
Vol 15 (04) ◽  
pp. 790-802 ◽  
Author(s):  
L. C. G. Rogers
Keyword(s):  
Convex Hull ◽  
Continuous Distribution ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Absolutely Continuous ◽  
Two Samples ◽  
Disjoint Convex ◽  
The Given

Suppose given an absolutely continuous distribution on the plane, and points P i, · · ·, Pj , Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj the convex hull of {P i, · · ·, Pj }, and by Γk the convex hull of (Πt, · · ·, Π k }, and writing pjk for the probability that Gj and Γ k are disjoint, certain properties of the array {pjk ; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p 1n ; n = 1,2, · · ·}, and asymptotic results for {pnn ; n = 1,2, · · ·}. Some examples are considered.

On resampling schemes for polytopes

2019 ◽  
Vol 56 (4) ◽  
pp. 959-980
Author(s):  
Weinan Qi ◽  
Mahmoud Zarepour
Keyword(s):  
Convex Hull ◽  
Asymptotic Theory ◽  
Convex Set ◽  
Real Life ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Underlying Distribution ◽  
Asymptotic Consistency ◽  
Computational Intractability ◽  
Practical Implications

AbstractThe convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. To approximate the distribution of the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately most of the asymptotic results are computationally intractable. To address this computational intractability, we consider consistent bootstrapping schemes for certain cases. Let $S_n=\{X_i\}_{i=1}^{n}$ be a sequence of independent and identically distributed random points uniformly distributed on an unknown convex set in $\mathbb{R}^{d}$ ($d\ge 2$ ). We suggest a bootstrapping scheme that relies on resampling uniformly from the convex hull of $S_n$ . Moreover, the resampling asymptotic consistency of certain functionals of convex hulls is derived under this bootstrapping scheme. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. For $d=2$ , we investigate the asymptotic consistency of the suggested bootstrapping scheme for the area of the symmetric difference and the perimeter difference between the actual convex set and its estimate. In all cases the consistency allows us to rely on the suggested resampling scheme to study the actual distributions, which are not computationally tractable.

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.

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).

Sequential selection of random vectors under a sum constraint

2004 ◽  
Vol 41 (1) ◽  
pp. 131-146
Author(s):  
Mario Stanke
Keyword(s):  
Continuous Distribution ◽  
Center Of Gravity ◽  
Continuous Measure ◽  
Random Vectors ◽  
Expected Number ◽  
Absolutely Continuous ◽  
Selection Policy ◽  
Sequential Selection ◽  
Absolutely Continuous Measure ◽  
Selection Of

We observe a sequence X1, X2,…, Xn of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the Xi we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1]d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section {x ∈ Q | 〈x, θ〉 ≤ 1}, where θ ∈ ℝd, θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

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.

Sequential selection of random vectors under a sum constraint

2004 ◽  
Vol 41 (01) ◽  
pp. 131-146
Author(s):  
Mario Stanke
Keyword(s):  
Continuous Distribution ◽  
Center Of Gravity ◽  
Continuous Measure ◽  
Random Vectors ◽  
Expected Number ◽  
Absolutely Continuous ◽  
Selection Policy ◽  
Sequential Selection ◽  
Absolutely Continuous Measure ◽  
Selection Of

We observe a sequence X 1, X 2,…, X n of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X i we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1] d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section { x ∈ Q | 〈 x , θ 〉 ≤ 1}, where θ ∈ ℝ d , θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

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