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.
Given 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 halfspaces 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.
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.
Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellation in $\mathbb{R}^d$. The distribution of the number of facets $f(Z)$ of the typical cell is investigated. It is shown, that under a well-spread condition on the directional distribution, the quantity $n^{\frac{2}{d-1}}\sqrt[n]{\mathbb{P}(f(Z)=n)}$ is bounded from above and from below. When $f(Z)$ is large, the isoperimetric ratio of $Z$ is bounded away from zero with high probability. These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of $Z$ and on the o…
Until now, little was known about properties of small cells in a Poisson hyperplane tessellation. The few existing results were either heuristic or applying only to the two dimensional case and for very specific size functionals and directional distributions. This paper fills this gap by providing a systematic study of small cells in a Poisson hyperplane tessellation of arbitrary dimension, arbitrary directional distribution $φ$ and with respect to an arbitrary size functional $Σ$. More precisely, we investigate the distribution of the typical cell $Z$, conditioned on the event $\{Σ(Z)<a\}$, w…
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 $K$.
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. Whi…
We study the asymptotic behavior of a size-marked point process of centers of large cells in a stationary and isotropic Poisson hyperplane mosaic in dimension $d \ge 2$. The sizes of the cells are measured by their inradius or their $k$th intrinsic volume ($k \ge 2$), for example. We prove a Poisson limit theorem for this process in Kantorovich-Rubinstein distance and thereby generalize a result in Chenavier and Hemsley (2016) in various directions. Our proof is based on a general Poisson process approximation result that extends a theorem in Bobrowski, Schulte and Yogeshwaran (2021).
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 infinity.
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…
This paper deals with the intersection point process of a stationary and isotropic Poisson hyperplane process in $\mathbb{R}^d$ of intensity $t>0$, where only hyperplanes that intersect a centred ball of radius $R>0$ are considered. Taking $R=t^{-\frac{d}{d+1}}$ it is shown that this point process converges in distribution, as $t\to\infty$, to a Poisson point process on $\mathbb{R}^d\setminus\{0\}$ whose intensity measure has power-law density proportional to $\|x\|^{-(d+1)}$ with respect to the Lebesgue measure. A bound on the speed of convergence in terms of the Kantorovich-Rubinstein distan…
A generalized version of a well-known problem of D. G. Kendall states that the zero cell of a stationary Poisson hyperplane tessellation in ${\mathbb{R}}^d$, under the condition that it has large volume, approximates with high probability a certain definite shape, which is determined by the directional distribution of the underlying hyperplane process. This result is extended here to typical $k$-faces of the tessellation, for $k\in\{2,...,d-1\}$. This requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction.
We consider the typical cell of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. It is well known that the expected vertex number of the typical cell is independent of the directional distribution of the hyperplane process. We give sharp bounds for the variance of this vertex number, showing, in particular, that the maximum of the variance is attained if and only if the distribution of the process is rotation invariant with respect to a suitable scalar product. The employed representation of the second moment of the vertex number is a special case of formulas prov…
We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in $\mathbb{R}^d$. This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab. 30 (1998) 640--656] for intersection points of motion-invariant Poisson line processes in $\mathbb{R}^2$. Our proof is based on Hoeffding's decomposition of $U$-statistics which seems to be more efficient and adequate to tackle the higher-dimensional case than the ``method of moments'' used in [Adv. in Appl. Probab. 30 (1998) 640--656] to treat the case $d=2$. Moreove…
Since the seminal work by Nagel and Weiss, the iteration stable (STIT) tessellations have attracted considerable interest in stochastic geometry as a natural and flexible, yet analytically tractable model for hierarchical spatial cell-splitting and crack-formation processes. We provide in this paper a fundamental link between typical characteristics of STIT tessellations and those of suitable mixtures of Poisson hyperplane tessellations using martingale techniques and general theory of piecewise deterministic Markov processes (PDMPs). As applications, new mean values and new distributional res…
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 …
We study random polytopes of the form $[X_1,\ldots,X_n]$ defined as convex hulls of independent and identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^d$ with one of the following densities: $$ f_{d,β} (x) = c_{d,β} (1-\|x\|^2)^β, \qquad \|x\| < 1, \quad \text{(beta distribution, $β>-1$)} $$ or $$ \tilde f_{d,β} (x) = \tilde{c}_{d,β} (1+\|x\|^2)^{-β}, \qquad x\in\mathbb{R}^d, \quad \text{(beta' distribution, $β>d/2$)}. $$ This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptot…
We consider the family of convex bodies obtained from an origin symmetric convex body $K$ by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call $K$-transform. In the special case, if $K$ is a segment not lying on any coordinate hyperplane, one obtains the family of zonoids and the cosine transform. In this case two facts are known: the vector sp…
In 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…
The isotropic constant $L_K$ is an affine-invariant measure of the spread of a convex body $K$. For a $d$-dimensional convex body $K$, $L_K$ can be defined by $L_K^{2d} = \det(A(K))/(\mathrm{vol}(K))^2$, where $A(K)$ is the covariance matrix of the uniform distribution on $K$. It is an outstanding open problem to find a tight asymptotic upper bound of the isotropic constant as a function of the dimension. It has been conjectured that there is a universal constant upper bound. The conjecture is known to be true for several families of bodies, in particular, highly symmetric bodies such as bodie…
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 $\sqrt{2π/3}\, d^{3/2}$, 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 π^…
A $U$-statistic of a Poisson point process is defined as the sum $\sum f(x_1,\ldots,x_k)$ over all (possibly infinitely many) $k$-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener-Itô chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for $U$-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investig…
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 $\mathbb R^d$. The expected $f$-vector is expressed through the coefficients of the polynomial $$ (1+ (d-1)^2x^2) (1+(d-3)^2 x^2) (1+(d-5)^2 x^2) \ldots. $$ 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 conve…
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 $\mathbb{R}^d$ 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…
Let $K \subset \R^d$ be a smooth convex set and let $¶_\la$ be a Poisson point process on $\R^d$ of intensity $\la$. The convex hull of $¶_\la \cap K$ is a random convex polytope $K_\la$. As $\la \to \infty$, we show that the variance of the number of $k$-dimensional faces of $K_\la$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$.
Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\to\infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ 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…
Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We achieve these results by first proving similar statements for weighted mean width approximations of convex bodies that admit a rollin…
This paper deals with the union set of a stationary Poisson process of cylinders in $\mathbb{R}^n$ having an $(n-m)$-dimensional base and an $m$-dimensional direction space, where $m\in\{0,1,\ldots,n-1\}$ and $n\geq 2$. 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 g…
We study the concentration of the norm of a random vector $Y$ uniformly sampled in the centered zero cell of two types of stationary and isotropic random mosaics in $\mathbb{R}^n$ for large dimensions $n$. For a stationary and isotropic Poisson-Voronoi mosaic, $Y$ has a radial and log-concave distribution, implying that ${|Y|}/{\mathbb{E}(|Y|^2)^{\frac{1}{2}}}$ approaches one for large $n$. Assuming the cell intensity of the random mosaic scales like $e^{n ρ_n}$, where $\lim_{n \to \infty} ρ_n = ρ$, $|Y|$ is on the order of $\sqrt{n}$ for large $n$. For the Poisson-Voronoi mosaic, we show that…
Let $η_t$ be a Poisson point process with intensity measure $tμ$, $t>0$, over a Borel space $\mathbb{X}$, 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 $t\to\infty$, 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 c…
In a previous work, two of the authors proposed a new proof of a well known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process. In this paper, we consider the particular case of the two-dimensional Boolean model where the grains are discs with random radii. We investigate the second-order term in this convergence when the Boolean model and the Poisson line process are coupled on the same probability space. A precise coupling between the Boolean model and the Poisson line process …
We present a model of a random field on a topological space $M$ that unifies well-known models such as the Poisson hyperplane tessellation model, the random token model, and the dead leaves model. In addition to generalizing these submodels from $\mathbb{R}^d$ to other spaces such as the $d$-dimensional unit sphere $\mathbb{S}^d$, our construction also extends the classical models themselves, e.g. by replacing the Poisson distribution by an arbitrary discrete distribution. Moreover, the method of construction directly produces an exact and fast simulation procedure. By investigating the covari…
We observe stationary random tessellations $X=\{Ξ_n\}_{n\ge1}$ in $\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their intersections with the $(d-1)$-facets of independent and identically distributed motion-invariant tessellations $X_n$ generated within each cell $Ξ_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In …
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gradient maps define novel investment strategies. The gradient maps may also be regarded as optimal transport maps on the simplex. In this paper we construct and study probability measures supported on spaces of concave functions. These measures may serve as prior distributions in …
