Random points in halfspheres

Bárány, Imre; Hug, Daniel; Reitzner, Matthias; Schneider, Rolf

doi:10.1002/rsa.20644

Cited by 39 publications

(83 citation statements)

References 18 publications

(27 reference statements)

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“…Thus the statement follows from Theorem 1.2 with ψ = ψ n and (11). Theorem 2.2 complements a recent result by Bárány, Hug, Reitzner and Schneider [2] for random polytopes in hemispheres.…”

Section: Spherical Spacesupporting

confidence: 71%

Weighted floating bodies and polytopal approximation

Besau

Ludwig

Werner

2018

Trans. Amer. Math. Soc.

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Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floating areas are combined to derive new asymptotic approximation results on the sphere, in hyperbolic space and in Hilbert geometries.2000 AMS subject classification: Primary 52A38; Secondary 52A27, 52A55, 53C60, 60D05.Let K be a convex body (that is, compact convex set) in R n . For δ > 0, the floating body K δ of K is obtained by cutting off all caps that have volume less or equal to δ. Extending results for smooth bodies (cf. [18]), Schütt and Werner [34] showed for a general convex bodywhere α n is an explicitly known positive constant (see Section 1.1). Here V n is n-dimensional volume, H n−1 (K, x) is the Gauss-Kronecker curvature at x ∈ ∂K and integration is with respect to the (n − 1)-dimensional Hausdorff measure. The integral on the right side is the affine surface area of K (cf. [20,22] and [31, Section 10.5] for more information). Affine surface area also determines the asymptotic behavior of random polytopes. Specifically, choose m points uniformly and independently in K and denote their convex hull by K m . The random polytope K m is easily seen to converge to K in the sense that E(V n (K) − V n (K m )) → 0 as m → ∞, where E denotes expectation. The asymptotic behavior of K m has been studied extensively since the 1960's, starting with the seminal results by Rényi and Sulanke [28, 29] (cf. [17, 27]). Extending results of Bárány [1], Schütt [33] was able to prove the analog to (1) for the random polytope K m in a general convex body K,

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Section: Spherical Spacesupporting

confidence: 71%

Weighted floating bodies and polytopal approximation

Besau

Ludwig

Werner

2018

Trans. Amer. Math. Soc.

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“…In addition, our weak limit theorem allows us to describe the expectation asymptotics of the conic intrinsic volumes (in fact, all three versions of them) of the induced random cone. This solves in an extended form a conjecture posed by Bárány, Hug, Reitzner and Schneider; see Section 9 in [5]. We also study separately the expected so-called T -functional of the convex hull of a general class of Poisson point processes in R d with a power-law intensity function x −(d+γ) ; see Theorem 2.12.…”

Section: Introductionmentioning

confidence: 87%

“…The paper is structured as follows. In Section 2.1 we first rephrase the relevant results from [5] and introduce the random convex cones for which various limit theorems are presented in Sections 2.2 and 2.3. Convex hulls of Poisson point processes with a power-law intensity function are the content of Section 2.4.…”

Section: Introductionmentioning

confidence: 99%

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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“…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%

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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Random points in halfspheres

Cited by 39 publications

References 18 publications

Weighted floating bodies and polytopal approximation

Weighted floating bodies and polytopal approximation

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

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