Related papers: Random points in halfspheres

We investigate the asymptotic properties of random polytopes arising as convex hulls of $n$ independent random points sampled from a family of block-beta distributions. Notably, this family includes the uniform distribution on a product of…

We consider random polytopes in the $d$-dimensional Euclidean space that are the convex hulls i.i.d. random points selected according to beta-prime distributions. These distributions are rotationally symmetric, heavy-tailed, and their…

We consider a stationary Poisson hyperplane process with given directional distribution and intensity in $d$-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body $K$ and consider the intersection…

Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I.…

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the $\ell_p$-unit sphere of $\mathbb R^n$ for some $1\leq p < \infty$ is considered. We prove that these random…

Let $K$ be a convex body in $\mathbb{R}^n$ and $f : \partial K \rightarrow \mathbb{R}_+$ a continuous, strictly positive function with $\int\limits_{\partial K} f(x) d \mu_{\partial K}(x) = 1$. We give an upper bound for the approximation…

Let $K\subset\mathbb S^{d-1}$ be a convex spherical body. Denote by $\Delta(K)$ the distance between two random points in $K$ and denote by $\sigma(K)$ the length of a random chord of $K$. We explicitly express the distribution of…

Given a stationary and isotropic Poisson hyperplane process and a convex body $K$ in ${\mathbb R}^d$, we consider the random polytope defined by the intersection of all closed halfspaces containing $K$ that are bounded by hyperplanes of the…

Let K be a convex set in R d and let K $\lambda$ be the convex hull of a homogeneous Poisson point process P $\lambda$ of intensity $\lambda$ on K. When K is a simple polytope, we establish scaling limits as $\lambda$ $\rightarrow$ $\infty$…

A simple symmetric random walk in the space $\mathbb{Z}^2$ is considered. The asymptotic behavior as the number of jumps tends to infinity of the probability that a fixed edge of the random walk lies in the polygon that forms the boundary…

A natural model for the approximation of a convex body $K$ in $\mathbb{R}^d$ by random polytopes is obtained as follows. Take a stationary Poisson hyperplane process in the space, and consider the random polytope $Z_K$ defined as the…

Choose n random, independent points in R^d according to a fixed distribution. The convex hull of these points is a random polytope. In some cases, central limit theorems have been proven for the components of f-vectors of random polytopes…

Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and…

We consider the moments of the volume of the symmetric convex hull of independent random points in an $n$-dimensional symmetric convex body. We calculate explicitly the second and fourth moments for $n$ points when the given body is $B_q^n$…

Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more…

Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of independent copies of a random vector $X$ in $\mathbb{R}^n$. We revisit the question to determine the asymptotic shape of the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$ where $N>n$. We…

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…

Let $\mu$ be a probability distribution on $\mathbb{R}^d$ which assigns measure zero to every hyperplane and $S$ a set of points sampled independently from $\mu$. What can be said about the expected combinatorial structure of the convex…

We study the Hausdorff distance between a random polytope, defined as the convex hull of i.i.d. random points, and the convex hull of the support of their distribution. As particular examples, we consider uniform distributions on convex…

In this paper we introduce a new sequence of quantities for random polytopes. Let $K_N=\conv\{X_1,...,X_N\}$ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body $K$ of $\R^n$. We…