Related papers: Random sections of spherical convex bodies

We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various…

In this paper we compare the different phenomena that occur when intersecting geometric objects with random geodesics on the unit sphere and inside convex bodies. On the high dimensional sphere we see that with probability bounded away from…

Langevin diffusion is a commonly used tool for sampling from a given distribution. In this work, we establish that when the target density $p^*$ is such that $\log p^*$ is $L$ smooth and $m$ strongly convex, discrete Langevin diffusion…

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…

A celebrated result of Beck shows that for any set of $N$ points on $\mathbb{S}^d$ there always exists a spherical cap $B \subset \mathbb{S}^d$ such that number of points in the cap deviates from the expected value $\sigma(B) \cdot N$ by at…

Let $K, D$ be $n$-dimensional convex bodes. Define the distance between $K$ and $D$ as $$ d(K,D) = \inf \{\lambda | T K \subset D+x \subset \lambda \cdot TK \}, $$ where the infimum is taken over all $x \in R^n$ and all invertible linear…

This technical note analyzes the properties of a random sequence of prolate hyperspheroids with common foci. Each prolate hyperspheroid in the sequence is defined by a sample drawn randomly from the previous volume such that the sample lies…

The probability distribution function (PDF) of the mass surface density of molecular clouds provides essential information about the structure of molecular cloud gas and condensed structures out of which stars may form. In general, the PDF…

We study random covers of a closed hyperbolic surface $\Sigma$, subject to the condition that, for $k\geq 2$, the fundamental group is isomorphic to the free group $F_k$. We show that asymptotically they distribute according to a specific…

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 $\lambda$ $\rightarrow$ $\infty$…

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We…

Motivated by a problem on the 67th William Lowell Putnam Mathematical Competition, we will summarize three different solutions found on a website. This Putman problem is a special case of Sylvester's four point problem! Suppose four points…

Consider an opaque medium which contains 3D particles. All particles are convex bodies of the same shape, but they vary in size. The particles are randomly positioned and oriented within the medium and cannot be observed directly. Taking a…

We establish a comprehensive probability theory for coherent transport of random waves through arbitrary linear media. The transmissivity distribution for random coherent waves is a fundamental B-spline with knots at the transmission…

The sub-keV mass range has long posed a challenge for the direct detection of dark matter via elastic scattering. In this Letter, we propose a new mechanism in which dark matter, assumed to be quadratically coupled to SM particles, scatters…

Motivated by models in engineering and also biology we determine in closed form the probability density function of the angle shaped by two random chords in a fixed disc. Our main result focus on the event in which the intersection locates…

In this paper, we provide explicit lower bounds with respect to some quantities of interest (parameters of the underlying distribution, dimension, geometrical characteristics of the domain, position of the origin, etc.) on the spectral gap…

The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the…

The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional sphere $S^d$ is called a lune. By the thickness of $L$ we mean the distance of the centers of the $(d-1)$-dimensional hemispheres bounding $L$.…

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according…