Related papers: Faces in random great hypersphere tessellations
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 study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a…
The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of $n$ independent standard normally distributed points in $\mathbb R^d$. We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point…
We prove several new results on the combinatorial structures of the unit spheres of the norms induced by Thurston's metric on the tangent and cotangent spaces of the Teichm{\"u}ller space of a closed surface of negative Euler…
We present two identities (contiguity relation and variation formula) concerning the volume of a spherically faced simplex in the Euclidean space. These identities are described in terms of Cayley-Menger determinants and their differentials…
This survey is devoted to recent developments in the statistical analysis of spherical data, with a view to applications in Cosmology. We will start from a brief discussion of Cosmological questions and motivations, arguing that most…
In this work we study a class of random convex sets that "interpolate" between polytopes and zonotopes. These sets arise from considering a $q^{th}$-moment ($q\geq 1$) of an average of order statistics of $1$-dimensional marginals of a…
We give a geometrically exact treatment of percolation through voids around assemblies of randomly placed impermeable barrier particles, introducing a computationally inexpensive approach to finding critical barrier density thresholds…
Understanding how particles are arranged on the sphere is not only central to numerous physical, biological, and materials systems but also finds applications in mathematics and in analysis of geophysical and meteorological measurements. In…
Consider randomly picked points inside the n-dimensional unit hypersphere centered at the origin of the Cartesian coordinate system. The Cartesian coordinates of the points are random variables, which form an n-dimensional vector for each…
The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects.…
A set of lower bounds on the continuum percolation threshold $\eta_c$ of overlapping convex hyperparticles of general nonspherical (anisotropic) shape with a specified orientational probability distribution in $d$-dimensional Euclidean…
We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a…
We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index $\alpha\in(1,2)$. When the number $n$ of…
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 $U_1,\ldots,U_n$…
We consider the situation when a globally defined four-dimensional field system is separated on two entangled sub-systems by a dynamical (random) two-dimensional surface. The reduced density matrix averaged over ensemble of random surfaces…
Clusters and groups of galaxies are highly aspherical, with shapes approximated by nearly prolate ellipsoids of revolution. An equally fundamental property is the shape of these objects in velocity space which is the anisotropy of the…
Three-dimensional random tessellations that are stable under iteration (STIT tessellations) are considered. They arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the…
Gaussian random polytopes have received a lot of attention especially in the case where the dimension is fixed and the number of points goes to infinity. Our focus is on the less studied case where the dimension goes to infinity and the…
The probability distribution function (PDF) of the mass surface density is an essential characteristic of the structure of molecular clouds or the interstellar medium in general. Observations of the PDF of molecular clouds indicate a…