scholarly journals On the approximation of a ball by random polytopes

1994 ◽  
Vol 26 (4) ◽  
pp. 876-892 ◽  
Author(s):  
K.-H. Küfer
Keyword(s):  
Convex Hull ◽  
Unit Ball ◽  
Surface Area ◽  
Limiting Distribution ◽  
Spherically Symmetric ◽  
Random Vectors ◽  
Symmetric Distributions ◽  
Random Polytopes ◽  
Random Polytope ◽  
Dimensional Unit

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a1,· ··, an. Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn)/Ε (Δ (Xn)) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.

scholarly journals On the approximation of a ball by random polytopes

1994 ◽  
Vol 26 (04) ◽  
pp. 876-892 ◽  
Author(s):  
K.-H. Küfer
Keyword(s):  
Convex Hull ◽  
Unit Ball ◽  
Surface Area ◽  
Limiting Distribution ◽  
Spherically Symmetric ◽  
Random Vectors ◽  
Symmetric Distributions ◽  
Random Polytopes ◽  
Random Polytope ◽  
Dimensional Unit

Letbe a sequence of independent and identically distributed random vectors drawn from thed-dimensional unit ballBdand letXnbe the random polytope generated as the convex hull ofa1,· ··,an.Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributedaiand2 we prove that the limiting distribution of Δ(Xn)/Ε(Δ(Xn)) forn→ ∞ (satisfies a 0–1 law. In particular, we show that Varforn→ ∞. We provide analogous results for spherically symmetric distributions inBdwith regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets ofXn.

Random polytopes in a ball

1984 ◽  
Vol 21 (4) ◽  
pp. 753-762 ◽  
Author(s):  
C. Buchta ◽  
J. Müller
Keyword(s):  
Convex Hull ◽  
Surface Area ◽  
Convex Polytope ◽  
Expected Number ◽  
Random Polytopes ◽  
Mean Width ◽  
Dimensional Ball

The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope. Its expected surface area, its expected mean width and its expected number of facets are explicitly determined.

Random polytopes in a ball

1984 ◽  
Vol 21 (04) ◽  
pp. 753-762 ◽  
Author(s):  
C. Buchta ◽  
J. Müller
Keyword(s):  
Convex Hull ◽  
Surface Area ◽  
Convex Polytope ◽  
Expected Number ◽  
Random Polytopes ◽  
Mean Width ◽  
Dimensional Ball

The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope. Its expected surface area, its expected mean width and its expected number of facets are explicitly determined.

scholarly journals Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

2008 ◽  
Vol 60 (1) ◽  
pp. 3-32 ◽  
Author(s):  
Károly Böröczky ◽  
Károly J. Böröczky ◽  
Carsten Schütt ◽  
Gergely Wintsche
Keyword(s):  
Unit Ball ◽  
Surface Area ◽  
Convex Bodies ◽  
Extreme Points ◽  
Thin Shells ◽  
Minimal Volume ◽  
Asymptotic Formulae ◽  
Mean Width

AbstractGiven r > 1, we consider convex bodies in En which contain a fixed unit ball, and whose extreme points are of distance at least r from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As r tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.

scholarly journals The maximum surface area polyhedron with five vertices inscribed in the sphere {\bb S}^{2}

2021 ◽  
Vol 77 (1) ◽  
pp. 67-74
Author(s):  
Jessica Donahue ◽  
Steven Hoehner ◽  
Ben Li
Keyword(s):  
Convex Hull ◽  
Surface Area ◽  
Coordination Polyhedron ◽  
Unit Sphere ◽  
The Other ◽  
Optimal Placement ◽  
The North ◽  
Trigonal Bipyramidal ◽  
Bipyramidal Structure ◽  
North And South

This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere {\bb S}^{2} so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices.

The convex hull of a spherically symmetric sample

1981 ◽  
Vol 13 (4) ◽  
pp. 751-763 ◽  
Author(s):  
William F. Eddy ◽  
James D. Gale
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Probability Measure ◽  
Convex Hull ◽  
Random Sample ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Extreme Value ◽  
Spherically Symmetric ◽  
Extreme Value Distributions

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.

scholarly journals Asymptotic shape of the convex hull of isotropic log-concave random vectors

2016 ◽  
Vol 75 ◽  
pp. 116-143 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Labrini Hioni ◽  
Antonis Tsolomitis
Keyword(s):  
Convex Hull ◽  
Random Vectors ◽  
Asymptotic Shape
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