Related papers: The polytopes in a Poisson hyperplane tessellation

Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye, we prove that the maximal $\infty$-step pro-nilfactor $X_\infty$ of a minimal system $(X,T)$ is the topological characteristic factor along polynomials in a certain sense.…

It is shown that rational extensions of the isotropic Dunkl oscillator in the plane can be obtained by adding some terms either to the radial equation or to the angular one obtained in the polar coordinates approach. In the former case, the…

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…

This paper deals with transportation polytopes in the probability simplex (that is, sets of categorical bivariate probability distributions with prescribed marginals). Information projections between such polytopes are studied, and a…

Poisson processes in the space of $k$-dimensional totally geodesic subspaces ($k$-flats) in a $d$-dimensional standard space of constant curvature $\kappa\in\{-1,0,1\}$ are studied, whose distributions are invariant under the isometries of…

Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the…

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d.…

With any max-stable random process $\eta$ on $\mathcal{X}=\mathbb{Z}^d$ or $\mathbb{R}^d$, we associate a random tessellation of the parameter space $\mathcal{X}$. The construction relies on the Poisson point process representation of the…

With any integer convex polytope $P\subset\midR^n$ we associate a multivariate hypergeometric polynomial whose set of exponents is $\midZ^{n}\cap P.$ This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic…

Let $A$ be any plane set. It is known that a holomorphic motion $h: A \times \mathbb{D} \to \mathbb{C}$ always extends to a holomorphic motion of the entire plane. It was recently shown that any isotopy $h: X \times [0,1] \to \mathbb{C}$,…

Consider a finite collection of affine hyperplanes in $\mathbb R^d$. The hyperplanes dissect $\mathbb R^d$ into finitely many polyhedral chambers. For a point $x\in \mathbb R^d$ and a chamber $P$ the metric projection of $x$ onto $P$ is the…

A variety of codimension $c$ in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension $c$. Positively hyperbolic hypersurfaces are defined by…

We provide a definitive classification of all finite sets of regular polygons that admit a tiling of the hyperbolic plane, thereby establishing the decidability of the Domino Problem for this class of prototiles. We show that admissibility…

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of finding…

We consider a stationary Poisson process of $k$-planes in the $d$-dimensional hyperbolic space $\mathbb H^d$ of constant curvature $-1$, with $d \ge 4$ and $1 \le k \le d-1$. It is known that, after centring and normalization, the total…

For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for…

We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of…

Assume $K$ is a convex body in $R^d$, and $X$ is a (large) finite subset of $K$. How many convex polytopes are there whose vertices come from $X$? What is the typical shape of such a polytope? How well the largest such polytope (which is…

We consider tessellations of the Euclidean $(d-1)$-sphere by $(d-2)$-dimensional great subspheres or, equivalently, tessellations of Euclidean $d$-space by hyperplanes through the origin; these we call conical tessellations. For random…

A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons…