Related papers: Poisson polytopes

The intrinsic volumes are measures of the content of a convex body. This paper uses probabilistic and information-theoretic methods to study the sequence of intrinsic volumes of a convex body. The main result states that the intrinsic…

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source…

Let $X_1,\ldots,X_n$ be independent random points in the closed unit ball of $\mathbb{R}^d$. Assume that each $X_i$ has a beta distribution with parameter $\beta_i \ge -1$: if $\beta_i>-1$, then $X_i$ has Lebesgue density proportional to…

We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a…

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $\beta_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil…

In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the non-empty subsets $A\subset \{1,..,n\}$, is it possible to construct a body $T\subset \mathbb{R}^n$ such that $x_A=|T_A|$ where $|T_A|$…

We prove dynamical local limits for the singular numbers of $p$-adic random matrix products at both the bulk and edge. The limit object which we construct, the reflecting Poisson sea, may thus be viewed as a $p$-adic analogue of line…

Given $\alpha \in (0, \infty)$ and $r \in (0, \infty)$, let ${\cal D}_{r, \alpha}$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha^2$. Consider the Poisson point process having uniform intensity density on ${\cal…

In this paper, we are interested in the behavior of the typical Poisson-Voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity. We prove a law of large numbers for…

Consider the random graph $G({\mathcal P}_{n},r)$ whose vertex set ${\mathcal P}_{n}$ is a Poisson point process of intensity $n$ on $(- \frac{1}{2}, \frac{1}{2}]^d$, $d \geq 2$. Any two vertices $X_i,X_j \in {\mathcal P}_{n}$ are connected…

In this note we discuss additional properties of mixed Poisson distributions. We discuss the convergence of mixed Poisson distributions to its mixing distribution for the scaling parameter tending to infinity. Moreover, we obtain a central…

We investigate the rate of convergence in the central limit theorem for convex sets. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the…

We derive entropy bounds for the absolute convex hull of vectors $X= (x_1 , \ldots , x_p)\in \mathbb{R}^{n \times p} $ in $\mathbb{R}^n$ and apply this to the case where $X$ is the $d$-fold tensor matrix $$X = \underbrace{\Psi \otimes…

For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(\eta_n)_{n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity…

A {\em convex hole} (or {\em empty convex polygon)} of a point set $P$ in the plane is a convex polygon with vertices in $P$, containing no points of $P$ in its interior. Let $R$ be a bounded convex region in the plane. We show that the…

Let $d$ be a nonnegative integer, and let $P \subset \mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\binom{2d+1}{d},$…

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In a previous article, we proved tight lower bounds for the coefficients of the generalized $h$-vector of a centrally symmetric rational polytope using intersection cohomology of the associated projective toric variety. Here we present a…

We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous…

Random union sets $Z$ associated with stationary Poisson processes of $k$-cylinders in $\mathbb{R}^d$ are considered. Under general conditions on the typical cylinder base a concentration inequality for the volume of $Z$ restricted to a…