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Matthias Reitzner

Publications and source records attributed to Matthias Reitzner.

32 records · Page 2Linked to original sources
arXiv2016

On the monotonicity of the moments of volumes of random simplices

In a $d$-dimensional convex body $K$ random points $X_0, \dots, X_d$ are chosen. Their convex hull is a random simplex. The expected volume of a random simplex is monotone under set inclusion, if $K \subset L$ implies that the expected volume of a random simplex in $K$ is smaller than the expected volume of a random simplex in $L$. Continuing work of Rademacher, it is shown that moments of the volume of random simplices are in general not monotone under set inclusion.

math.MG↗
arXiv2015

Random points in halfspheres

A random spherical polytope $P_n$ in a spherically convex set $K \subset S^d$ as considered here is the spherical convex hull of $n$ independent, uniformly distributed random points in $K$. The behaviour of $P_n$ for a spherically convex set $K$ contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when $K$ is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as $n$ tends to infinity, of the expectation of several characteristics of $P_n$, such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates.

math.PR↗
arXiv2015

Concentration for Poisson U-Statistics: Subgraph Counts in Random Geometric Graphs

Concentration inequalities for subgraph counts in random geometric graphs built over Poisson point processes are proved. The estimates give upper bounds for the probabilities $\mathbb{P}(N\geq M +r)$ and $\mathbb{P}(N\leq M - r)$ where $M$ is either a median or the expectation of a subgraph count $N$. The bounds for the lower tail have a fast Gaussian decay and the bounds for the upper tail satisfy an optimality condition. A special feature of the presented inequalities is that the underlying Poisson process does not need to have finite intensity measure. The tail estimates for subgraph counts follow from concentration inequalities for more general local Poisson U-statistics. These bounds are proved using recent general concentration results for Poisson U-statistics and techniques based on the convex distance for Poisson point processes.

math.PR↗
arXiv2015

U-statistics in stochastic geometry

This survey will appear as a chapter of the forthcoming book [19]. A U-statistic of order $k$ with kernel $f:\X^k \to \R^d$ over a Poisson process is defined in \cite{ReiSch11} as$$ \sum\_{x\_1, \dots , x\_k \in η^k\_{\neq}} f(x\_1, \dots, x\_k) $$ under appropriate integrability assumptions on $f$. U-statistics play an important role in stochastic geometry since many interesting functionals can be written as U-statistics, like intrinsic volumes of intersection processes, characteristics of random geometric graphs, volumes of random simplices, and many others, see for instance \cite{ LacPec13, LPST,ReiSch11}. It turns out that the Wiener-Ito chaos expansion of a U-statistic is finite and thus Malliavin calculus is a particularly suitable method. Variance estimates, the approximation of the covariance structure and limit theorems which have been out of reach for many years can be derived. In this chapter we state the fundamental properties of U-statistics and investigate moment formulae. The main object of the chapter is to introduce the available limit theorems.

math.PR↗
arXiv2015

Poisson polyhedra in high dimensions

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.

math.MG↗
arXiv2014

Beyond the Efron-Buchta identities: distributional results for Poisson polytopes

Let $Π$ be a random polytope defined as the convex hull of the points of a Poisson point process. Identities involving the moment generating function of the measure of $Π$, the number of vertices of $Π$ and the number of non-vertices of $Π$ are proven. Equivalently, identities for higher moments of the mentioned random variables are given. This generalizes analogous identities for functionals of convex hulls of i.i.d points by Efron and Buchta.

math.PR↗
arXiv2013

Central limit theorems for $U$-statistics of Poisson point processes

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 investigated.

math.PR↗
arXiv2012

The monotonicity of f-vectors of random polytopes

Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is increasing in n. In dimension d>=3 we prove that if lim(E((f[d -1](Kn))/(An^c)->1 when n->infinity for some constants A and c > 0 then the function E(f[d-1](Kn)) is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.

cs.CG↗
arXiv2012

Many empty triangles have a common edge

Given a finite point set $X$ in the plane, the degree of a pair $\{x,y\} \subset X$ is the number of empty triangles $t=conv\{x,y,z\}$, where empty means $t\cap X=\{x,y,z\}$. Define $deg X$ as the maximal degree of a pair in $X$. Our main result is that if $X$ is a random sample of $n$ independent and uniform points from a fixed convex body, then $deg X \ge cn/\ln n$ in expectation.

math.PR↗
arXiv2011

Set Reconstruction by Voronoi cells

For a Borel set $A$ and a homogeneous Poisson point process $η$ in $\R^d$ of intensity $λ>0$, define the Poisson--Voronoi approximation $ A_η$ of $A$ as a union of all Voronoi cells with nuclei from $η$ lying in $A$. If $A$ has a finite volume and perimeter we find an exact asymptotic of $\E\Vol(AΔA_η)$ as $λ\to\infty$ where $\Vol$ is the Lebesgue measure. Estimates for all moments of $\Vol(A_η)$ and $\Vol(AΔA_η)$ together with their asymptotics for large $λ$ are obtained as well.

math.PR↗
arXiv2010

Poisson polytopes

We prove the central limit theorem for the volume and the $f$-vector of the Poisson random polytope $Π_η$ in a fixed convex polytope $P\subset\mathbb{R}^d$. Here, $Π_η$ is the convex hull of the intersection of a Poisson process $X$ of intensity $η$ with $P$.

math.PR↗
arXiv2009

Poisson--Voronoi approximation

Let $X$ be a Poisson point process and $K\subset\mathbb{R}^d$ a measurable set. Construct the Voronoi cells of all points $x\in X$ with respect to $X$, and denote by $v_X(K)$ the union of all Voronoi cells with nucleus in $K$. For $K$ a compact convex set the expectation of the volume difference $V(v_X(K))-V(K)$ and the symmetric difference $V(v_X(K)\triangle K)$ is computed. Precise estimates for the variance of both quantities are obtained which follow from a new jackknife inequality for the variance of functionals of a Poisson point process. Concentration inequalities for both quantities are proved using Azuma's inequality.

math.PR↗
arXiv2006

Elementary moves on triangulations

It is proved that a triangulation of a polyhedron can always be transformed into any other triangulation of the polyhedron by using only elementary moves. One consequence is that an additive function (valuation) defined only on simplices may always be extended to an additive function on all polyhedra.

math.MG↗