Related papers: Beta Polytopes and Beta Cones: An Exactly Solvable…

Motivated by applications in Bayesian analysis we introduce a multidimensional beta distribution in an ordered simplex. We study properties of this distribution and connect them with the generalized incomplete beta function. This function…

We prove the central limit theorem for the volume and the $f$-vector of the Poisson random polytope $\Pi_{\eta}$ in a fixed convex polytope $P\subset\mathbb{R}^d$. Here, $\Pi_{\eta}$ is the convex hull of the intersection of a Poisson…

We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans…

The region of entropic vectors is a convex cone that has been shown to be at the core of many fundamental limits for problems in multiterminal data compression, network coding, and multimedia transmission. This cone has been shown to be…

We consider the problem of estimating the mean vector $\theta$ of a $d$-dimensional spherically symmetric distributed $X$ based on balanced loss functions of the forms: {\bf (i)} $\omega \rho(\|\de-\de_{0}\|^{2}) +(1-\omega)\rho(\|\de -…

In this paper two new classes of stationary random simplicial tessellations, the so-called $\beta$- and $\beta'$-Delaunay tessellations, are introduced. Their construction is based on a space-time paraboloid hull process and generalizes…

The collection $\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\mathbb{R}^{\binom{n}{2}}$, known as the metric polytope. In this paper, we study the…

Let $P_1,\dots, P_n$ and $Q_1,\dots, Q_n$ be convex polytopes in $\mathbb{R}^n$ such that $P_i\subset Q_i$. It is well-known that the mixed volume has the monotonicity property: $V(P_1,\dots,P_n)\leq V(Q_1,\dots,Q_n)$. We give two criteria…

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 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 $B_q^n$…

The non-elementary integrals $\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\beta\ge1,\alpha\le\beta+1$ and $\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1,…

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…

Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $\beta=(\beta(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates.…

We present an explicit closed-form formula for the vertices of the classical cut polytope $\operatorname{CUT}(n)$, defined as the convex hull of cut vectors of the complete graph $K_n$. Our derivation proceeds via a related polytope,…

Conical zeta values associated with rational convex polyhedral cones generalise multiple zeta values. We renormalise conical zeta values at poles by means of a generalisation of Connes and Kreimer's Algebraic Birkhoff Factorisation. This…

We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the…

We define an n-dimensional polytope Pi_n(x), depending on parameters x_i>0, whose combinatorial properties are closely connected with empirical distributions, plane trees, plane partitions, parking functions, and the associahedron. In…

Some special functions are particularly relevant in applied probability and statistics. For example, the incomplete beta function is the cumulative central beta distribution. In this paper, we consider the inversion of the central…

We describe a maximum entropy approach for computing volumes and counting integer points in polyhedra. To estimate the number of points from a particular set X in R^n in a polyhedron P in R^n, by solving a certain entropy maximization…

The intrinsic volumes are measures of the content of a convex body. This paper uses probabilistic and information-theoretic methods to study the sequence of intrinsic volumes of a convex body. The main result states that the intrinsic…