Poisson Hyperplane Research Articles

Extremal behavior of large cells in the Poisson hyperplane mosaic

Extremal behavior of large cells in the Poisson hyperplane mosaic

Analysis of heterogeneous Markov media for particle transport problems.

Markov media provide a prototype class of stochastic geometries that are widely used in order to model several complex and disordered systems encompassing, e.g., turbulent fluids and plasma, atmospheric layers, or biological tissues, especially in relation to particle transport problems. In several key applications, the statistical properties of random media may display spatial gradients due to material stratification, which means that the typical spatial scale and the probability of finding a given material phase at a spatial location become nonhomogeneous. In this paper we investigate the main features of spatially heterogeneous Markov media, using Poisson hyperplane tessellations and Arak polygonal fields. We show that both models can generate geometry realizations sharing Markov-like properties, and discuss their distinct advantages and drawbacks in terms of flexibility and ease of use. The impact of these models on the observables related to particle transport will be assessed using Monte Carlo simulations.

Face numbers of high-dimensional Poisson zero cells

Let Z d \mathcal Z_d be the zero cell of a d d -dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of k k -dimensional faces of Z d \mathcal Z_d , as d → ∞ d\to \infty . For example, we show that the expected number of hyperfaces of Z d \mathcal Z_d is asymptotically equivalent to 2 π / 3 d 3 / 2 \sqrt {2\pi /3}\, d^{3/2} , as d → ∞ d\to \infty . We also prove that the expected solid angle of a random cone spanned by d d random vectors that are independent and uniformly distributed on the unit upper half-sphere in R d \mathbb R^{d} is asymptotic to 3 π − d \sqrt 3 \pi ^{-d} , as d → ∞ d\to \infty .

On strongly rigid hyperfluctuating random measures

AbstractIn contrast to previous belief, we provide examples of stationary ergodic random measures that are both hyperfluctuating and strongly rigid. Therefore we study hyperplane intersection processes (HIPs) that are formed by the vertices of Poisson hyperplane tessellations. These HIPs are known to be hyperfluctuating, that is, the variance of the number of points in a bounded observation window grows faster than the size of the window. Here we show that the HIPs exhibit a particularly strong rigidity property. For any bounded Borel set B, an exponentially small (bounded) stopping set suffices to reconstruct the position of all points in B and, in fact, all hyperplanes intersecting B. Therefore the random measures supported by the hyperplane intersections of arbitrary (but fixed) dimension, are also hyperfluctuating. Our examples aid the search for relations between correlations, density fluctuations, and rigidity properties.

Weak convergence of the intersection point process of Poisson hyperplanes

Cet article traite du processus ponctuel d’intersection d’un processus d’hyperplans de Poisson stationnaire et isotrope dans Rd d’intensité t>0, où seuls les hyperplans qui intersectent une boule centrée de rayon R>0 sont considérés. En prenant R=t−dd+1, on montre que ce processus ponctuel converge en distribution, lorsque t→∞, vers un processus ponctuel de Poisson dans Rd∖{0} dont la mesure d’intensité a une densité proportionnelle à ‖x‖−(d+1) par rapport à la mesure de Lebesgue. Une borne sur la vitesse de convergence en termes de distance de Kantorovich–Rubinstein est également fournie. En arrière-plan se trouve un théorème général d’approximation de Poisson fonctionnel sur les espaces de Poisson abstraits. Les implications sur la convergence faible de l’enveloppe convexe du processus de point d’intersection et la convergence de son vecteur f sont également discutées, réfutant et corrigeant ainsi une conjecture de Devroye et Toussaint (J. Algorithms 14 (1993) 381–394) en géométrie computationelle.

Beta-star polytopes and hyperbolic stochastic geometry

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 Rd 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.

Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

This paper deals with the union set of a stationary Poisson process of cylinders in Rn having an (n−m)-dimensional base and an m-dimensional direction space, where m∈{0,1,…,n−1} and n≥2. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or m-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of geometric functionals, including the intrinsic volumes. In this context the asymptotic variance constant is analysed in detail, which in contrast to the Boolean model leads to a new degeneracy phenomenon. A quantitative central limit theory is developed in a multivariate set-up as well.

Facial structure of strongly convex sets generated by random samples

The K-hull of a compact set A⊆Rd, where K⊆Rd is a fixed compact convex body, is the intersection of all translates of K that contain A. A set is called K-strongly convex if it coincides with its K-hull. We propose a general approach to the analysis of facial structure of K-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of k-dimensional faces, for all k=0,…,d−1. We then apply our theory in the case when A=Ξn is a sample of n points picked uniformly at random from K. We show that in this case the set of x∈Rd such that x+K contains the sample Ξn, upon multiplying by n, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From these results we deduce convergence in distribution of the corresponding f-vector of the K-hull of Ξn to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the f-vector.

The polytopes in a Poisson hyperplane tessellation

For a stationary Poisson hyperplane tessellation $X$ in ${\mathbb R}^d$, whose directional distribution satisfies some mild conditions (which hold in the isotropic case, for example), it was recently shown that with probability one every combinatorial type of a simple $d$-polytope is realized infinitely often by the polytopes of $X$. This result is strengthened here: with probability one, every such combinatorial type appears among the polytopes of $X$ not only infinitely often, but with positive density.

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Poisson processes in the space of (d-1)-dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature -1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all dge 2, it is shown that in case (ii) the central limit theorem holds for din {2,3} and fails if dge 4 and k=d-1 or if dge 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

EXPECTEDf‐VECTOR OF THE POISSON ZERO POLYTOPE AND RANDOM CONVEX HULLS IN THE HALF‐SPHERE

We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in R d . The expected f-vector is expressed through the coefficients of the polynomial ( 1 + ( d − 1 ) 2 x 2 ) ( 1 + ( d − 3 ) 2 x 2 ) ( 1 + ( d − 5 ) 2 x 2 ) … . Also, we compute explicitly the expected f-vector and the expected volume of the spherical convex hull of n random points sampled uniformly and independently from the d-dimensional half-sphere. In the case when n = d + 2 , we compute the probability that this spherical convex hull is a spherical simplex, thus solving the half-sphere analogue of the Sylvester four-point problem.

A new approach to weak convergence of random cones and polytopes

AbstractA new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation, generated bynindependent and uniformly distributed random linear hyperplanes in$\mathbb {R}^{d+1}$, weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in$\mathbb {R}^d$, as$n\to \infty $.

Beta polytopes and Poisson polyhedra: f-vectors and angles

We study random polytopes of the form [X1,…,Xn] defined as convex hulls of independent and identically distributed random points X1,…,Xn in Rd with one of the following densities:fd,β(x)=cd,β(1−‖x‖2)β,‖x‖<1,(beta distribution, β>−1) orf˜d,β(x)=c˜d,β(1+‖x‖2)−β,x∈Rd,(beta' distribution, β>d/2). Here, cd,β and c˜d,β are normalizing constants. This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptotic formulae for the expected number of k-faces of [X1,…,Xn] for arbitrary k∈{0,1,…,d−1}. We prove that for any such k this expected number is strictly monotonically increasing with n. Also, we compute the expected internal and external angles of these polytopes at faces of every dimension and, more generally, the expected conic intrinsic volumes of their tangent cones. By passing to the large n limit in the beta' case, we compute the expected f-vector of the convex hull of Poisson point processes with power-law intensity function. Using convex duality, we derive exact formulae for the expected number of k-faces of the zero cell for a class of isotropic Poisson hyperplane tessellations in Rd. This family includes the zero cell of a classical stationary and isotropic Poisson hyperplane tessellation and the typical cell of a stationary Poisson–Voronoi tessellation as special cases. In addition, we prove precise limit theorems for this f-vector in the high-dimensional regime, as d→∞. Finally, we relate the d-dimensional beta and beta' distributions to the generalized Pareto distributions known in extreme-value theory.

POISSON HYPERPLANE PROCESSES AND APPROXIMATION OF CONVEX BODIES

A natural model for the approximation of a convex body $K$ in $\mathbb{R}^d$ by random polytopes is obtained as follows. Take a stationary Poisson hyperplane process in the space, and consider the random polytope $Z_K$ defined as the intersection of all closed halfspaces containing $K$ that are bounded by hyperplanes of the process not intersecting $K$. If $f$ is a functional on convex bodies, then for increasing intensities of the process, the expectation of the difference $f(Z_K)-f(K)$ may or may not converge to zero. If it does, then the order of convergence and possible limit relations are of interest. We study these questions if $f$ is either the hitting functional or the mean width.

Interaction of Poisson hyperplane processes and convex bodies

AbstractGiven a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

Small faces in stationary Poisson hyperplane tessellations

AbstractWe consider the tessellation induced by a stationary Poisson hyperplane process in d‐dimensional Euclidean space. Under a suitable assumption on the directional distribution, and measuring the k‐faces of the tessellation by a suitable size functional, we determine a limit distribution for the shape of the typical k‐face, under the condition of given size and this tending to zero. The limit distribution is concentrated on simplices. This extends a result of Gilles Bonnet.

The stochastic geometry of unconstrained one-bit data compression

A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of $\mathbb{R} ^{n}$ or a stationary Poisson point process in $\mathbb{R} ^{n}$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension $n$ to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressed sensing and in source coding.

Chord length distribution in d-dimensional anisotropic Markov media

Markov media are often used as a prototype model in the analysis of linear particle transport in disordered materials. For this class of stochastic geometries, it is assumed that the chord lengths must follow an exponential distribution, with a direction-dependent average if anisotropy effects are to be taken into account. The practical realizability of Markov media in arbitrary dimension has been a long-standing open question. In this work we show that Poisson hyperplane tessellations provide an explicit construction for random media satisfying the Markov property and easily including anisotropy. The average chord length can be computed explicitly and is be shown to be intimately related to the statistical properties of the tessellation cells and in particular to their surface-to-volume ratio. A computer code that is able to generate anisotropic Poisson tessellations in arbitrary dimension restricted to a given finite domain is developed, and the convergence to exact asymptotic formulas for the chord length distribution and the polyhedral features of the tessellation cells is established by extensive Monte Carlo simulations in the limit of domains having an infinite size.

Regenerative processes for Poisson zero polytopes

Abstract Let (Mt:t&gt;0) be a Markov process of tessellations of ℝℓ, and let (𝒞t:t&gt;0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at𝒞at:t∈ℝ),a&gt;1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.

Concentration and moderate deviations for Poisson polytopes and polyhedra

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 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.