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Circles of a single size can pack together densely in a hexagonal lattice, but adding in size variety disrupts the order of those packings. We conduct simulations which generate dense random packings of circles with specified size…

For a random field on a general discrete set, we introduce a condition that the range of the correlation from each site is within a predefined compact set D. For such a random field omega defined on the model set Lambda that satisfies a…

Let $f_n$ be a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon $P_n$ of $n$ sides. Suppose ${\mathcal T}_n$ is a random triangulation, sampled uniformly out of all possible…

Using the LePage representation, a strictly stable random element in a Banach space with $\alpha\in(0,2)$ can be represented as a sum of points of a Poisson process. This point process is union-stable, i.e. the union of its two independent…

The existence of the weak limit as n --> infinity of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random…

Let $\mathbb{P}_{\kappa}(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_\kappa$, a regular $\kappa$-gon with area $1$, are in convex position, that is, form the vertex set of a…

Inspired by the works of Zagier, we study the probability measures $\nu(t)$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle…

This paper is motivated by the following observation. Take a 3 x 3 random (Haar distributed) orthogonal matrix $\Gamma$, and use it to "rotate" the north pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point $u=\Gamma x_0$…

Pick $d+1$ points uniformly at random on the unit sphere in $\mathbb R^d$. What is the expected value of the angle sum of the simplex spanned by these points? Choose $n$ points uniformly at random in the $d$-dimensional ball. What is the…

Plant differently colored points in the plane, then let random points ("Poisson rain") fall, and give each new point the color of the nearest existing point. Previous investigation and simulations strongly suggest that the colored regions…

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

We give an effective method to compute the entropy for polynomials orthogonal on a segment of the real axis that uses as input data only the coefficients of the recurrence relation satisfied by these polynomials. This algorithm is based on…

This paper deals with the typical cell in a Poisson line tessellation in the plane whose directional distribution is concentrated on three equally spread values with possibly different weights. Such a random polygon can only be a triangle,…

We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of…

The polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. The class includes the uniform and trapezoidal distributions, and is an alternative to the beta…

The study of extremal problems on triangle areas was initiated in a series of papers by Erd\H{o}s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that…

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the…

We establish a relationship between the two important central lines of the triangle, the Euler line and the Brocard axis, in a configuration with an arbitrary rectangle and a random point. The classical Cartesian coordinate system method…

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently…

We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…