Convex Hulls of Uniform Samples from a Convex Polygon

Groeneboom, Piet

doi:10.1239/aap/1339878714

Cited by 13 publications

(2 citation statements)

References 11 publications

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“…For this case some mathematical results are known [37][38][39]. In particular the remaining areã A n = 1 − A(n) outside the convex hull is considered.…”

Section: A Independent Pointsmentioning

confidence: 99%

Convex hulls of random walks: Large-deviation properties

2015

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We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter L and the mean area A have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P (A) and P (L). In this paper, we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P (A) and P (L) are as small as 10 −300 . We analyse (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T , a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/ √ T , respecively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

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“…For this case some mathematical results are known [37][38][39]. In particular the remaining areã A n = 1 − A(n) outside the convex hull is considered.…”

Section: A Independent Pointsmentioning

confidence: 99%

Convex hulls of random walks: Large-deviation properties

2015

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“…The approach used in this paper is a modification of the methods proposed by [4,5,12] and adapted to a wider class of initial distributions.…”

Section: Introductionmentioning

confidence: 99%

On Limit Theorem for the Number of Vertices of the Convex Hulls in a Unit Disk

Khamdamov

Хамдамов²

2020

Journal of Siberian Federal University. Mathematics &Amp; Physics

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This paper is devoted to further investigation of the property of a number of vertices of convex hulls generated by independent observations of a two-dimensional random vector with regular distributions near the boundary of support when it is a unit disk. Following P. Groeneboom [4], the Binomial point process is approximated by the Poisson point process near the boundary of support and vertex processes of convex hulls are constructed. The properties of strong mixing and martingality of vertex processes are investigated. Using these properties, asymptotic expressions are obtained for the expectations and variance of the vertex processes that correspond to the results previously obtained by H. Carnal [2]. Further, using the properties of strong mixing of vertex processes, the central limit theorem for a number of vertices of a convex hull is proved

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The Duality of the Volumes and the Numbers of Vertices of Random Polytopes

Buchta

2023

Discrete Comput Geom

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An identity due to Efron dating from 1965 relates the expected volume of the convex hull of n random points to the expected number of vertices of the convex hull of $$n+1$$ n + 1 random points. Forty years later this identity was extended from expected values to higher moments. The generalized identity has attracted considerable interest. Whereas the left-hand side of the generalized identity—concerning the volume—has an immediate geometric interpretation, this is not the case for the right-hand side—concerning the number of vertices. A transformation of the right-hand side applying an identity for elementary symmetric polynomials overcomes the blemish. The arising formula reveals a duality between the volumes and the numbers of vertices of random polytopes.

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Convex Hulls of Uniform Samples from a Convex Polygon

Cited by 13 publications

References 11 publications

Convex hulls of random walks: Large-deviation properties

Convex hulls of random walks: Large-deviation properties

On Limit Theorem for the Number of Vertices of the Convex Hulls in a Unit Disk

The Duality of the Volumes and the Numbers of Vertices of Random Polytopes

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