On the boundary structure of the convex hull of random points

The Gauss-Minkowski correspondence in R 2 states the existence of a homeomorphism between the probability measures µ on [0, 2π] such that 2π 0 e ix dµ(x) = 0 and the compact convex sets (CCS) of the plane with perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS -for example, the Minkowski sum -have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notion of convolution of CCS. Additionally, we give a proof that a polygonal curve associated with a sample of n random variables (satisfying 2π 0 e ix dµ(x) = 0) converges to a CCS associated with µ at speed √ n, a result much similar to the convergence of the empirical process in statistics. Finally, we employ this correspondence to present models of smooth random CCS and simulations.

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“…As a consequence of Lebesgue's dominated convergence theorem, A(µ k ) converges to the right hand side of (7).…”

Section: Convolution Of Measures / Convolution Of Ccsmentioning

confidence: 96%

“…This is the so-called Brunn-Minkowski inequality; it implies that A(λµ+(1−λ)ν) ≥ min{A(µ), A(ν)}. It can be proved using Hurwitz formula (7) and the Cauchy-Schwarz inequality.…”

Section: Mixtures Of Ccs / Minkowski Summentioning

confidence: 98%

Compact convex sets of the plane and probability theory

Marckert

Renault

2014

ESAIM: PS

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“…37].) The same approach can be used to obtain higher moments of N n ; however, the technical details are more involved and are therefore elaborated in a separate paper [9].…”

Section: The Variance Of the Number Of Verticesmentioning

confidence: 99%

Exact Formulae for Variances of Functionals of Convex Hulls

Buchta

2013

Advances in Applied Probability

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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).

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“…This is the subject of the current research by Buchta. Results for higher moments of the convex hull of a uniform sample from a triangle with vertices (0, 0), (0, 1), and (1, 0) are given in Buchta (2012). Let N(a, b) and D(a, b) be defined as in Theorem 3.1.…”

Section: Convex Hullsmentioning

confidence: 99%

“…I would like to posthumously thank Tomasz Schreiber for sending me the unpublished preprint Nagaev and Khamdamov (1991), and Christian Buchta for making me aware of Buchta (2003) and sending me the preprint Buchta (2012).…”

Section: Acknowledgementsmentioning

confidence: 99%

Convex Hulls of Uniform Samples from a Convex Polygon

Groeneboom

2012

Advances in Applied Probability

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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 r log n}/{ 10 27 r log n} 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).

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On the boundary structure of the convex hull of random points

Cited by 11 publications

References 18 publications

Compact convex sets of the plane and probability theory

Compact convex sets of the plane and probability theory

Exact Formulae for Variances of Functionals of Convex Hulls

Convex Hulls of Uniform Samples from a Convex Polygon

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