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

2014 ◽  
Vol 46 (3) ◽  
pp. 622-642 ◽  
Author(s):  
Julia Hörrmann ◽  
Daniel Hug
Keyword(s):  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Explicit Formula ◽  
Dimensional Euclidean Space ◽  
Arbitrary Dimensions ◽  
Zero Cell ◽  
Parametric Class

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to ∞.

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

2014 ◽  
Vol 46 (03) ◽  
pp. 622-642 ◽  
Author(s):  
Julia Hörrmann ◽  
Daniel Hug
Keyword(s):  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Explicit Formula ◽  
Dimensional Euclidean Space ◽  
Arbitrary Dimensions ◽  
Zero Cell ◽  
Parametric Class

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to ∞.

scholarly journals Approximation Properties of Random Polytopes Associated with Poisson Hyperplane Processes

2014 ◽  
Vol 46 (4) ◽  
pp. 919-936
Author(s):  
Daniel Hug ◽  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Hausdorff Distance ◽  
Dimensional Euclidean Space ◽  
Approximation Properties ◽  
Random Polytopes ◽  
Directional Distribution ◽  
Zero Cell

We consider a stationary Poisson hyperplane process with given directional distribution and intensity in d-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body K and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing K. We study how well these random polytopes approximate K (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of K.

scholarly journals Weighted faces of Poisson hyperplane tessellations

2009 ◽  
Vol 41 (03) ◽  
pp. 682-694 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Euclidean Space ◽  
Upper Bounds ◽  
Dimensional Euclidean Space ◽  
Lower And Upper Bounds ◽  
Intrinsic Volumes ◽  
Vertex Number ◽  
Zero Cell ◽  
Lower Dimensional

We study lower-dimensional volume-weighted typical faces of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. After showing how their distribution can be derived from that of the zero cell, we obtain sharp lower and upper bounds for the expected vertex number of the volume-weighted typical k-face (k=2,…,d). The bounds are respectively attained by parallel mosaics and by isotropic tessellations. We conclude with a remark on expected face numbers and expected intrinsic volumes of the zero cell.

scholarly journals Faces with given directions in anisotropic Poisson hyperplane mosaics

2011 ◽  
Vol 43 (2) ◽  
pp. 308-321 ◽  
Author(s):  
Daniel Hug ◽  
Rolf Schneider
Keyword(s):  
Euclidean Space ◽  
Crucial Role ◽  
Conditional Distribution ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Different Types ◽  
Typical Cell ◽  
Zero Cell ◽  
Asymptotic Problem

For stationary Poisson hyperplane tessellations in d-dimensional Euclidean space and a dimension k ∈ {1, …, d}, we investigate the typical k-face and the weighted typical k-face (weighted by k-dimensional volume), without isotropy assumptions on the tessellation. The case k = d concerns the previously studied typical cell and zero cell, respectively. For k < d, we first find the conditional distribution of the typical k-face or weighted typical k-face, given its direction. Then we investigate how the shapes of the faces are influenced by assumptions of different types: either via containment of convex bodies of given volume (including a new result for k = d), or, for weighted typical k-faces, in the spirit of D. G. Kendall's asymptotic problem, suitably generalized. In all these results on typical or weighted typical k-faces with given direction space L, the Blaschke body of the section process of the underlying hyperplane process with L plays a crucial role.

scholarly journals Central Limit Theorems for Volume and Surface Content of Stationary Poisson Cylinder Processes in Expanding Domains

2013 ◽  
Vol 45 (02) ◽  
pp. 312-331
Author(s):  
Lothar Heinrich ◽  
Malte Spiess
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Limit Theorems ◽  
Dimensional Euclidean Space ◽  
Surface Content ◽  
Directional Distributions ◽  
Shaped Domain

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d−1) which are dilated by independent and identically distributed random compact cylinder bases taken from the corresponding (d−k)-dimensional orthogonal complement. If the second moment of the (d−k)-volume of the typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k) th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.

Weighted faces of Poisson hyperplane tessellations

2009 ◽  
Vol 41 (3) ◽  
pp. 682-694 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Euclidean Space ◽  
Upper Bounds ◽  
Dimensional Euclidean Space ◽  
Lower And Upper Bounds ◽  
Intrinsic Volumes ◽  
Vertex Number ◽  
Zero Cell ◽  
Lower Dimensional

We study lower-dimensional volume-weighted typical faces of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. After showing how their distribution can be derived from that of the zero cell, we obtain sharp lower and upper bounds for the expected vertex number of the volume-weighted typical k-face (k=2,…,d). The bounds are respectively attained by parallel mosaics and by isotropic tessellations. We conclude with a remark on expected face numbers and expected intrinsic volumes of the zero cell.

scholarly journals Approximation Properties of Random Polytopes Associated with Poisson Hyperplane Processes

2014 ◽  
Vol 46 (04) ◽  
pp. 919-936 ◽  
Author(s):  
Daniel Hug ◽  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Hausdorff Distance ◽  
Dimensional Euclidean Space ◽  
Approximation Properties ◽  
Random Polytopes ◽  
Directional Distribution ◽  
Zero Cell

We consider a stationary Poisson hyperplane process with given directional distribution and intensity ind-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex bodyKand consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containingK. We study how well these random polytopes approximateK(measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties ofK.

scholarly journals On the Asymptotic Behaviour of a Diffusion Process with Singular Drift

1975 ◽  
Vol 57 ◽  
pp. 87-106
Author(s):  
Hitoshi Kaneta
Keyword(s):  
Brownian Motion ◽  
Markov Process ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Asymptotic Behaviour ◽  
Markov Processes ◽  
Smooth Boundary ◽  
Dimensional Euclidean Space ◽  
Singular Drift ◽  
Strong Markov

We discuss some peculiar features of the diffusion process whose characterization is given below. Let D be a bounded domain in the d-dimensional Euclidean space Ed with a smooth boundary ∂D. The domain D contains open balls (i = 1, · · ·, n) which are mutually disjoint. Our process is a diffusion process on the state space D ∪ ∂D which is locally equivalent to the Brownian motion except on the spheres ∂ and the boundary ∂D. By a diffusion process we mean a continuous strong Markov process. As to the terminology about Markov processes we refer to [2].

scholarly journals Faces with given directions in anisotropic Poisson hyperplane mosaics

2011 ◽  
Vol 43 (02) ◽  
pp. 308-321 ◽  
Author(s):  
Daniel Hug ◽  
Rolf Schneider
Keyword(s):  
Euclidean Space ◽  
Crucial Role ◽  
Conditional Distribution ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Different Types ◽  
Typical Cell ◽  
Zero Cell ◽  
Asymptotic Problem

For stationary Poisson hyperplane tessellations in d-dimensional Euclidean space and a dimension k ∈ {1, …, d}, we investigate the typical k-face and the weighted typical k-face (weighted by k-dimensional volume), without isotropy assumptions on the tessellation. The case k = d concerns the previously studied typical cell and zero cell, respectively. For k &lt; d, we first find the conditional distribution of the typical k-face or weighted typical k-face, given its direction. Then we investigate how the shapes of the faces are influenced by assumptions of different types: either via containment of convex bodies of given volume (including a new result for k = d), or, for weighted typical k-faces, in the spirit of D. G. Kendall's asymptotic problem, suitably generalized. In all these results on typical or weighted typical k-faces with given direction space L, the Blaschke body of the section process of the underlying hyperplane process with L plays a crucial role.

scholarly journals Central Limit Theorems for Volume and Surface Content of Stationary Poisson Cylinder Processes in Expanding Domains

2013 ◽  
Vol 45 (2) ◽  
pp. 312-331 ◽  
Author(s):  
Lothar Heinrich ◽  
Malte Spiess
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Limit Theorems ◽  
Dimensional Euclidean Space ◽  
Surface Content ◽  
Directional Distributions ◽  
Shaped Domain

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d−1) which are dilated by independent and identically distributed random compact cylinder bases taken from the corresponding (d−k)-dimensional orthogonal complement. If the second moment of the (d−k)-volume of the typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k) th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.

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