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

Bräker, H.; Hsing, Tailen; Bingham, Ν. H.

doi:10.1239/aap/1035228070

Cited by 17 publications

(6 citation statements)

References 21 publications

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“…The asymptotic distribution of the Hausdorff distance between a planar convex body K and K n has been determined with high precision by Bräker, Hsing, and Bingham in [28]. A general central limit theorem was proved by Matthias Reitzner in [51].…”

Section: Central Limit Theoremsmentioning

confidence: 99%

Random points and lattice points in convex bodies

Bárány

2008

Bull. Amer. Math. Soc.

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Abstract. Assume K ⊂ R d is a convex body and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong to X? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? We are interested in these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester's famous four-point problem and from the theory of random polytopes. The second case is when X = K ∩ Z d where Z d is the lattice of integer points in R d and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar. Sylvester's four-point problemThe study of random points in convex bodies started with an innocent looking question. The year was 1864. The place was London. The journal was the Educational Times. Problem 1941 came from J. J. Sylvester [60]. It read: "Show that the chance of four points forming the apices of a reentrant quadrilateral is 1/4 if they be taken at random in an indefinite plane." Several answers came in. Most of them were different. In 1865 Sylvester [61] concluded, "This problem does not admit of a determinate solution." The reason is, as we all know by now, in "at random in an indefinite plane", since there is no natural probability measure on it. Sylvester immediately modified the question: Let K be a convex body in the plane and choose four random, independent, and uniform points from K. What is the probability that they form the vertices of a reentrant quadrilateral, or, in recent terminology, that their convex hull is a triangle. Further, for what K is this probability the smallest and the largest.This question has become known as Sylvester's four-point problem, and it has proved to be extremely fertile. It took more than fifty years (and several erroneous proofs) to find the answer. Blaschke showed in [26] and [27] that the probability in question is largest for a triangle and smallest for the disk (or any ellipse). His solution used the method of symmetrization and "shaking down" that have become standard tools since.Sylvester's question, and its subsequent solution, determined the direction of research for a long time. Many papers have been written starting with the setting: Let X n = {x 1 , ..., x n } be a random, independent, uniform sample of n points from

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Section: Central Limit Theoremsmentioning

confidence: 99%

Random points and lattice points in convex bodies

Bárány

2008

Bull. Amer. Math. Soc.

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“…For results on asymptotic distributions, see, e.g. the papers of Bräker et al [5], who computed the Hausdorff distance between a convex set and the convex hull of an inner random sample, and Molchanov [12] on plug-in estimation of level sets.…”

Section: For Definitions and Properties) In This Equation Given A Smentioning

confidence: 99%

Nonparametric estimation of boundary measures and related functionals: asymptotic results

2009

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We study a nonparametric method for estimating the boundary measure of a compact body G ⊂ R d (the boundary length when d = 2 and the surface area for d = 3) in the case when this measure agrees with the corresponding Minkowski content. The estimator we consider is closely related to the one proposed in Cuevas, Fraiman and Rodríguez-Casal (2007). Our method relies on two sets of random points, drawn inside and outside the set G, with different sampling intensities. Strong consistency and asymptotic normality are obtained under some shape hypotheses on the set G. Some applications and practical aspects are briefly discussed.

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“…in distribution, again, with more or less explicit constants c 1 and c 2 . The asymptotic distribution of the Hausdorff distance between a planar convex body K and K n has been determined with high precision by Bräker, Hsing, and Bingham in [28]. A general central limit theorem was proved by Matthias Reitzner in [51].…”

Section: Central Limit Theoremsmentioning

confidence: 99%

Untitled

2008

Bull. Amer. Math. Soc.

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Assume K ⊂ R d is a convex body and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong to X? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? We are interested in these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester's famous four-point problem and from the theory of random polytopes. The second case is when X = K ∩ Z d where Z d is the lattice of integer points in R d and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar.

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

Cited by 17 publications

References 21 publications

Random points and lattice points in convex bodies

Random points and lattice points in convex bodies

Nonparametric estimation of boundary measures and related functionals: asymptotic results

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