On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

Hoerrmann, Julia; Hug, Daniel

doi:10.1017/s0001867800007291

Cited by 7 publications

(11 citation statements)

References 24 publications

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“…This interpretation is one of the motivations for studying properties of Z 0 ∩ L, where L is an m-dimensional linear subspace of R n . Thanks to the special structure of our tessellation model, we are able to show a transfer principle, which allows us to translate results from Z 0 to intersections Z 0 ∩ L of Z 0 with a subspace L. Combined with a recent result from [18] this yields Theorem 1.1.…”

Section: Introductionmentioning

confidence: 78%

Poisson polyhedra in high dimensions

Hörrmann

Hug

Reitzner

et al. 2015

Advances in Mathematics

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The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f -vector of the zero cell to certain dual intrinsic volumes.

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Section: Introductionmentioning

confidence: 78%

Poisson polyhedra in high dimensions

Hörrmann

Hug

Reitzner

et al. 2015

Advances in Mathematics

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“…If r = d, Z 0 is equal in distribution to the so-called typical cell of a Poisson-Voronoi tessellation as considered in the previous section, see [28]. Thus, the tessellation induced by η t interpolates in some sense between the translation-invariant Poisson hyperplane and the Poisson-Voronoi tessellation, which explains the recent interest in this model [10,11,13]. For more background material about random tessellations (and in particular Poisson hyperplane and Poisson-Voronoi tessellations) we refer to Chapter 10 in [28] and Chapter 9 in [7].…”

Section: Hyperplane Tessellationsmentioning

confidence: 91%

Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

Schulte

Thäle

2016

Stochastic Analysis for Poisson Point Processes

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Let η t be a Poisson point process with intensity measure tµ, t > 0, over a Borel space X, where µ is a fixed measure. Another point process ξ t on the real line is constructed by applying a symmetric function f to every k-tuple of distinct points of η t . It is shown that ξ t behaves after appropriate rescaling like a Poisson point process, as t → ∞, under suitable conditions on η t and f . This also implies Weibull limit theorems for related extreme values. The result is then applied to investigate problems arising in stochastic geometry, including small cells in Voronoi tessellations, random simplices generated by non-stationary hyperplane processes, triangular counts with angular constraints and non-intersecting k-flats. Similar results are derived if the underlying Poisson point process is replaced by a binomial point process.

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“…This parametric family of random polyhedra has attracted considerable interest in recent years because of its connections to high-dimensional convex geometry and to a version of the famous problem of D.G. Kendall asking for the asymptotic geometry of 'large' mosaic cells, see [13,14,23,24,28,39]. It includes the following special case that has received particular attention and is well known in the literature, cf.…”

Section: Applications To Poisson Polyhedramentioning

confidence: 99%

Concentration and moderate deviations for Poisson polytopes and polyhedra

Grote¹,

Thäle²

2018

Bernoulli

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The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the d-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to deduce, by duality, fine probabilistic estimates and moderate deviation principles for combinatorial parameters of a class of zero cells associated with Poisson hyperplane mosaics. As a special case this comprises the typical Poisson-Voronoi cell conditioned on having large inradius.

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On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

Cited by 7 publications

References 24 publications

Poisson polyhedra in high dimensions

Poisson polyhedra in high dimensions

Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

Concentration and moderate deviations for Poisson polytopes and polyhedra

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