On the area and perimeter of a random convex hull in a bounded convex set

1998 ◽  
Vol 111 (4) ◽  
pp. 517-550 ◽  
Author(s):  
H. Bräker ◽  
T. Hsing
Keyword(s):  
Convex Hull ◽  
Convex Set ◽  
Area And Perimeter ◽  
Bounded Convex ◽  
Random Convex

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

Convergence in mean of some characteristics of the convex hull

1989 ◽  
Vol 21 (3) ◽  
pp. 526-542 ◽  
Author(s):  
Henk Brozius
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Regular Variation ◽  
Poisson Point Process ◽  
Random Vectors ◽  
Area And Perimeter ◽  
Convergence In Mean ◽  
Quantities Of Interest ◽  
Independent And Identically Distributed

A sequence Xn, 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of {X1, …, Xn} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

On the LLN for the number of vertices of a random convex hull

2000 ◽  
Vol 32 (03) ◽  
pp. 675-681 ◽  
Author(s):  
Bruno Massé
Keyword(s):  
Convex Hull ◽  
Law Of Large Numbers ◽  
Large Numbers ◽  
General Conjecture ◽  
Common Parent ◽  
Random Convex

For several common parent laws, the number of vertices of a sample convex hull follows a kind of law of large numbers. We exhibit an example of a parent law which contradicts a general conjecture about this matter.

scholarly journals A Convex Cover for Closed Unit Curves has Area at Least 0.0975

2021 ◽  
pp. 2050006
Author(s):  
Bogdan Grechuk ◽  
Sittichoke Som-am
Keyword(s):  
Lower Bound ◽  
Convex Hull ◽  
Convex Set ◽  
Search Algorithm ◽  
Unit Length ◽  
Plane Curve ◽  
Geometric Methods ◽  
Convex Cover ◽  
Minimal Area ◽  
Closed Plane

We combine geometric methods with a numerical box search algorithm to show that the minimal area of a convex set in the plane which can cover every closed plane curve of unit length is at least [Formula: see text]. This improves the best previous lower bound of [Formula: see text]. In fact, we show that the minimal area of the convex hull of circle, equilateral triangle, and rectangle of perimeter [Formula: see text] is between [Formula: see text] and [Formula: see text].

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 Area and Perimeter of the Convex Hull of Stochastic Points

2016 ◽  
Vol 59 (8) ◽  
pp. 1144-1154 ◽  
Author(s):  
Pablo Pérez-Lantero
Keyword(s):  
Convex Hull ◽  
Area And Perimeter

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.

On the Hausdorff distance between a convex set and an interior random convex hull

1998 ◽  
Vol 30 (2) ◽  
pp. 295-316 ◽  
Author(s):  
H. Bräker ◽  
T. Hsing ◽  
N. H. Bingham
Keyword(s):  
Convex Hull ◽  
Hausdorff Distance ◽  
Convex Set ◽  
Compact Convex ◽  
Decisive Role ◽  
Value Theory ◽  
Limit Laws ◽  
First Case ◽  
Double Exponential ◽  
Compact Convex Set

The problem of estimating an unknown compact convex set K in the plane, from a sample (X1,···,Xn) of points independently and uniformly distributed over K, is considered. Let Kn be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that(Δn - bn)/an → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

Coverage problems and random convex hulls

1982 ◽  
Vol 19 (3) ◽  
pp. 546-561 ◽  
Author(s):  
Nicholas P. Jewell ◽  
Joseph P. Romano
Keyword(s):  
Convex Hull ◽  
Finite Number ◽  
Random Sample ◽  
Bivariate Distribution ◽  
Convex Hulls ◽  
Coverage Problem ◽  
Coverage Problems ◽  
Random Convex

Consider the placement of a finite number of arcs on the circle of circumference 2π where the midpoint and length of each arc follows an arbitrary bivariate distribution. In the case where each arc has lengthπ, the probability that the circle is completely covered is equal to the probability that the convex hull of a finite random sample of points, chosen according to a certain bivariate distribution in the plane contains the origin. In general, we show that evaluating the probability that the random convex hull contains a fixed disc is equivalent to solving the general coverage problem where the midpoint and length of each arc follows an arbitrary bivariate distribution. Exact formulae for the above probabilities are obtained and some examples are considered.

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