Monotonicity of facet numbers of random convex hulls

Bonnet, Gilles; Grote, Julian; Temesvari, Daniel; Thäle, Christoph; Turchi, Nicola; Wespi, Florian

doi:10.1016/j.jmaa.2017.06.054

Cited by 21 publications

(30 citation statements)

References 17 publications

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“…The next proposition describes the density according to which these points are distributed. The result is a consequence of [6,Proposition 4.2] and, in a more general set-up, has been proved in the argument of [7,Theorem 7].…”

Section: The Weak Convergence Theoremmentioning

confidence: 78%

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Kabluchko

Marynych

Temesvari

et al. 2019

Probab. Theory Relat. Fields

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Let U 1 , U 2 , . . . be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as n → ∞, the f -vector of the (d + 1)-dimensional convex cone C n generated by U 1 , . . . , U n weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f -vector of C n and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of C n can be expressed through the expected f -vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone C n weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to x −(d+γ) , where γ = 1. We compute the expected number of facets, the expected intrinsic volumes and the expected T -functional of this random convex hull for arbitrary γ > 0.

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Section: The Weak Convergence Theoremmentioning

confidence: 78%

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Kabluchko

Marynych

Temesvari

et al. 2019

Probab. Theory Relat. Fields

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“…For

α > - 1

we consider the distribution on

S_{+}^{d}

whose density with respect to the uniform distribution on

S_{+}^{d}

is given by

{\overset{f}{true}}_{d, α} false(x false) = c_{d, α} x_{d + 1}^{α}, x = (x_{1}, \dots, x_{d + 1}) \in {double-struckS}_{+}^{d} .

Here,

c_{d, α}

is a normalizing constant. The spherical convex hull

{\hat{P}}_{n, d}^{α}

of n independent random points on

S_{+}^{d}

distributed according to the density

{\hat{f}}_{d, α}

has been studied in (for general α) and (for the special case

α = 0

). In particular, it has been shown in [, Section 5] that the expected number of facets of the spherical random polytope

{\hat{P}}_{n, d}^{α}

coincides with that of

{\tilde{P}}_{n, d}^{β}

for the choice

β = \frac{1}{2} false(α + d + 1 false)

.…”

Section: Special Casesmentioning

confidence: 99%

“…The spherical convex hull

{\hat{P}}_{n, d}^{α}

of n independent random points on

S_{+}^{d}

distributed according to the density

{\hat{f}}_{d, α}

has been studied in (for general α) and (for the special case

α = 0

). In particular, it has been shown in [, Section 5] that the expected number of facets of the spherical random polytope

{\hat{P}}_{n, d}^{α}

coincides with that of

{\tilde{P}}_{n, d}^{β}

for the choice

β = \frac{1}{2} false(α + d + 1 false)

. Thus, also yields a new and explicit formula for the expected number of facets of spherical convex hull

{\hat{P}}_{n, d}^{α}

.…”

Section: Special Casesmentioning

confidence: 99%

Expected intrinsic volumes and facet numbers of random beta‐polytopes

Kabluchko

Temesvari

Thäle

2018

Mathematische Nachrichten

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Let X1,⋯,Xn be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: fd,βfalse(xfalse)=const·()1−∥x∥2β,false∥xfalse∥<1,(thebetacase)and truef∼d,βfalse(xfalse)=const·()1+∥x∥2−β,x∈double-struckRd,(thebeta"case).We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of X1,⋯,Xn. Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when β↓−1, respectively β↑+∞, we can also cover the models in which X1,⋯,Xn are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of ±X1,⋯,±Xn and 0,X1,⋯,Xn. One of the main tools used in the proofs is the Blaschke–Petkantschin formula.

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“…Asymptotic estimates on the expected volume of the polytope P β N,n , as N → ∞, were derived by Affentranger [1] for any fixed dimension n and parameter β. Note also, that the gnomonic projection of a uniformly distributed point on the half-sphere is beta prime distributed, which is exploited in [5] and [12].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Threshold phenomena for high-dimensional random polytopes

Bonnet

Chasapis

Grote

et al. 2019

Commun. Contemp. Math.

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Let X 1 , . . . , X N , N > n, be independent random points in R n , distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension n tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.

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Monotonicity of facet numbers of random convex hulls

Cited by 21 publications

References 17 publications

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Expected intrinsic volumes and facet numbers of random beta‐polytopes

Threshold phenomena for high-dimensional random polytopes

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