Polytopes with Vertices Chosen Randomly from the Boundary of a Convex Body

Schütt, Carsten; Werner, Elisabeth M.

doi:10.1007/978-3-540-36428-3_19

Cited by 93 publications

(159 citation statements)

References 32 publications

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“…Theorems 2.3 and 2.4 are similar to a result of J. S. Müller, in [4], about approximation of the Euclidean ball by random polytopes (see [5] and the references cited therein for related results and a discussion of similar questions of approximation). In our notation, Müller's result is an asymptotic formula for the difference vol n (B n 2 ) − E vol n (L N ).…”

Section: 3supporting

confidence: 66%

“…The asymptotic treatment in [4], however, is for the case when n is fixed and N → ∞. A major extension of Müller's result was done by Schütt and Werner in [5]. Namely, let K be a convex body whose boundary satisfies certain regularity conditions.…”

Section: 3mentioning

confidence: 99%

“…Namely, let K be a convex body whose boundary satisfies certain regularity conditions. Let C N be the convex hull of N points chosen randomly from the boundary of K. The authors of [5] derive an asymptotic formula for vol n (K) − E vol n (C N ), where, as in [4], n is fixed and N → ∞.…”

Section: 3mentioning

confidence: 99%

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Volume thresholds for Gaussian and spherical random polytopes and their duals

Pivovarov

2007

Studia Math.

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Abstract. Let g be a Gaussian random vector in R n . Let N = N (n) be a positive integer and let K N be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumesFor a large range of R = R(n), we establish a sharp threshold for N , above which V N → 1 as n → ∞, and below which V N → 0 as n → ∞. We also consider the case when K N is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both R ∈ (0, 1) and R = 1. Lastly, we prove complementary results for polytopes generated by random facets.

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Section: 3supporting

confidence: 66%

Section: 3mentioning

confidence: 99%

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Volume thresholds for Gaussian and spherical random polytopes and their duals

Pivovarov

2007

Studia Math.

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“…During the past decade various elements of the L p Brunn-Minkowski theory have attracted increased attention (see e.g. [3], [4], [5], [8], [9] [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [22], [24], [25], [26], [27], [28], [29]). …”

mentioning

confidence: 99%

On the Lp Minkowski Problem for Polytopes

Hug

Lutwak²,

Yang³

et al. 2005

Discrete Comput Geom

186

147

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Abstract. Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data L p Minkowski problem for all p > 1.As observed by Schneider [23], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R d , with that of Minkowski combinations of convex bodies. One of the cornerstones of the Brunn-Minkowski theory is the classical Minkowski problem. For polytopes the problem asks for the necessary and sufficient conditions on a set of unit vectors u 1 , . . . , u n ∈ S d−1 and a set of real numbers α 1 , . . . , α n > 0 that guarantee the existence of a polytope, P , in R d with n facets whose outer unit normals are u 1 , . . . , u n and such that the facet whose outer unit normal is u i has area (i.e.,

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“…The random sample may come from the normal distribution, or from the boundary of K. Here a recent result of Schütt and Werner [54] should be mentioned. In a long and intricate proof they show the precise asymptotic behaviour of vol(K \ K n ) when K is a smooth convex body and the random points are chosen from the boundary of K according to some probability distribution.…”

Section: Further Resultsmentioning

confidence: 99%

Random Polytopes, Convex Bodies, and Approximation

Weil¹

Stochastic Geometry

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Assume K ⊂ R d is a convex body and X n ⊂ K is a random sample of n uniform, independent points from K. The convex hull of X n is a convex polytope K n called random polytope inscribed in K. We are going to investigate various properties of this polytope: for instance how well it approximates K, or how many vertices and facets it has. It turns out that K n is very close to the so called floating body inscribed in K with parameter 1/n. To show this we develop and use the technique of cap coverings and Macbeath regions. Its power will be illustrated, besides random polytopes, on several examples: floating bodies, lattice polytopes, and approximation problems.

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Polytopes with Vertices Chosen Randomly from the Boundary of a Convex Body

Cited by 93 publications

References 32 publications

Volume thresholds for Gaussian and spherical random polytopes and their duals

Volume thresholds for Gaussian and spherical random polytopes and their duals

On the Lp Minkowski Problem for Polytopes

Random Polytopes, Convex Bodies, and Approximation

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