11 results on '"Kabluchko, Zakhar"'
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2. The β-Delaunay tessellation IV: Mixing properties and central limit theorems.
- Author
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Gusakova, Anna, Kabluchko, Zakhar, and Thäle, Christoph
- Subjects
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CENTRAL limit theorem , *LIMIT theorems , *STOCHASTIC geometry - Abstract
Various mixing properties of β -, β ′ - and Gaussian-Delaunay tessellations in ℝ d − 1 are studied. It is shown that these tessellation models are absolutely regular, or β -mixing. In the β - and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the β ′ -case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of β - and Gaussian-Delaunay tessellations are established. This includes the number of k -dimensional faces and the k -volume of the k -skeleton for k ∈ { 0 , 1 , ... , d − 1 }. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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3. Face numbers of high-dimensional Poisson zero cells.
- Author
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Kabluchko, Zakhar
- Subjects
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STOCHASTIC geometry , *CONES , *TESSELLATIONS (Mathematics) , *ANGLES - Abstract
Let \mathcal Z_d be the zero cell of a d-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of k-dimensional faces of \mathcal Z_d, as d\to \infty. For example, we show that the expected number of hyperfaces of \mathcal Z_d is asymptotically equivalent to \sqrt {2\pi /3}\, d^{3/2}, as d\to \infty. We also prove that the expected solid angle of a random cone spanned by d random vectors that are independent and uniformly distributed on the unit upper half-sphere in \mathbb R^{d} is asymptotic to \sqrt 3 \pi ^{-d}, as d\to \infty. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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4. The $\beta$ -Delaunay tessellation: Description of the model and geometry of typical cells.
- Author
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Gusakova, Anna, Kabluchko, Zakhar, and Thäle, Christoph
- Subjects
GEOMETRIC modeling ,TESSELLATIONS (Mathematics) ,PARABOLOID ,STOCHASTIC geometry ,POISSON processes ,POINT processes - Abstract
In this paper we introduce two new classes of stationary random simplicial tessellations, the so-called $\beta$ - and $\beta^{\prime}$ -Delaunay tessellations. Their construction is based on a space–time paraboloid hull process and generalizes that of the classical Poisson–Delaunay tessellation. We explicitly identify the distribution of volume-power-weighted typical cells, establishing thereby a remarkable connection to the classes of $\beta$ - and $\beta^{\prime}$ -polytopes. These representations are used to determine the principal characteristics of such cells, including volume moments, expected angle sums, and cell intensities. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
5. Conic intrinsic volumes of Weyl chambers.
- Author
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Godland, Thomas and Kabluchko, Zakhar
- Subjects
STOCHASTIC geometry ,CONIC sections ,RANDOM walks ,ANGLES - Abstract
A new, direct proof of the formulas for the conic intrinsic volumes of the Weyl chambers of types A
n-1 , Bn and Dn is given. These formulas express the conic intrinsic volumes in terms of the Stirling numbers of the first kind and their B- and D-analogues. The proof involves an explicit determination of the internal and external angles of the faces of the Weyl chambers. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF
6. Convex cones spanned by regular polytopes.
- Author
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Kabluchko, Zakhar and Seidel, Hauke
- Subjects
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STOCHASTIC geometry , *POLYTOPES , *CONES , *INTERSECTION numbers , *CUBES , *ANGLES - Abstract
We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
7. The β-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions.
- Author
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Gusakova, Anna, Kabluchko, Zakhar, and Thäle, Christoph
- Subjects
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TESSELLATIONS (Mathematics) , *GENERALIZATION , *POISSON algebras , *CENTRAL limit theorem , *DEVIATION (Statistics) - Abstract
The β-Delaunay tessellation in ℝd-1 is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a βDelaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex inℝd-1. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional β-Delaunay tessellation is analysed, as d → ∞. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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8. The Typical Cell of a Voronoi Tessellation on the Sphere.
- Author
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Kabluchko, Zakhar and Thäle, Christoph
- Subjects
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SPHERES , *TESSELLATIONS (Mathematics) , *BETA distribution , *POLYTOPES , *STOCHASTIC geometry - Abstract
The typical cell of a Voronoi tessellation generated by n + 1 uniformly distributed random points on the d-dimensional unit sphere S d is studied. Its f-vector is identified in distribution with the f-vector of a beta' polytope generated by n random points in R d . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases d ∈ { 2 , 3 , 4 } are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
9. Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes.
- Author
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Kabluchko, Zakhar
- Subjects
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POLYTOPES , *STOCHASTIC geometry , *CONVEX bodies , *GAMMA functions , *PROBLEM solving - Abstract
Consider a random simplex [ X 1 , ... , X n ] defined as the convex hull of independent identically distributed (i.i.d.) random points X 1 , ... , X n in R n - 1 with the following beta density: Let J n , k (β) be the expected internal angle of the simplex [ X 1 , ... , X n ] at its face [ X 1 , ... , X k ] . Define J ~ n , k (β) analogously for i.i.d. random points distributed according to the beta ′ density f ~ n - 1 , β (x) ∝ (1 + ‖ x ‖ 2 ) - β , x ∈ R n - 1 , β > (n - 1) / 2. We derive formulae for J n , k (β) and J ~ n , k (β) which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of β . For J n , 1 (± 1 / 2) we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope K n , d : = [ U 1 , ... , U n ] where U 1 , ... , U n are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of K n , d : lim n → ∞ n - (d - 1) / (d + 1) E f (K n , d) = c d · Ω (K) , where Ω (K) is the affine surface area of K, and c d is an unknown vector not depending on K. We compute c d explicitly in dimensions up to d = 10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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10. Conical tessellations associated with Weyl chambers.
- Author
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Godland, Thomas and Kabluchko, Zakhar
- Subjects
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STOCHASTIC geometry , *TESSELLATIONS (Mathematics) , *HYPERPLANES , *CONES , *FUNCTIONALS - Abstract
We consider d-dimensional random vectors Y1, . . . ,Yn that satisfy a mild general position assumption a.s. The hyperplanes (Yi−Yj)⊥ (1 ≤ i < j ≤ n) generate a conical tessellation of the Euclidean d-space which is closely related to the Weyl chambers of type An−1. We determine the number of cones in this tessellation and show that it is a.s. constant. For a random cone chosen uniformly at random from this random tessellation, we compute expectations of several geometric functionals. These include the face numbers, as well as the conic intrinsic volumes and the conical quermassintegrals. Under the additional assumption of exchangeability on Y1, . . . ,Yn, the same is done for the dual random cones which have the same distribution as the positive hull of Y1−Y2, . . . , Yn−1−Yn conditioned on the event that this positive hull is not equal to Rd. All these expectations turn out to be distribution-free. Similarly, we consider the conical tessellation induced by the hyperplanes (Yi + Yj)⊥ (1 ≤ i < j ≤ n), (Yi − Yj)⊥ (1 ≤ i < j ≤ n), Yi⊥ (1 ≤ i ≤ n). This tessellation is closely related to the Weyl chambers of type Bn. We compute the number of cones in this tessellation and the expectations of various geometric functionals for random cones drawn from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected non-trivially by a certain linear subspace in general position. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
11. Beta-star polytopes and hyperbolic stochastic geometry.
- Author
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Godland, Thomas, Kabluchko, Zakhar, and Thäle, Christoph
- Subjects
- *
STOCHASTIC geometry , *HYPERBOLIC geometry , *HYPERBOLIC spaces , *UNIT ball (Mathematics) , *POISSON processes , *POLYTOPES , *STOCHASTIC matrices - Abstract
Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in R d with density proportional to (‖ x ‖ 2 − 1) − β , where ‖ x ‖ > 1 and β > d / 2. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of k -dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected f -vector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
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