Related papers: A refinement of the Sylvester problem: Probabiliti…

Given $d+1$ sets of points, or colours, $S_1,\ldots,S_{d+1}$ in $\mathbb R^d$, a colourful simplex is a set $T\subseteq\bigcup_{i=1}^{d+1}S_i$ such that $|T\cap S_i|\leq 1$, for all $i\in\{1,\ldots,d+1\}$. The colourful Carath\'eodory…

We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$-gon with a small disk in the centre. This model provides convenient and uniform interpretations for…

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.

This article discusses the problem of determining whether a given point, or set of points, lies within the convex hull of another set of points in $d$ dimensions. This problem arises naturally in a statistical context when using a…

Let $X$ be the mosaic generated by a stationary Poisson hyperplane process $\hat X$ in ${\mathbb R}^d$. Under some mild conditions on the spherical directional distribution of $\hat X$ (which are satisfied, for example, if the process is…

This paper establishes an analogue of the Robinson--Schensted correspondence for cylindric tableaux. In particular, for any pair of positive integers $(d,L)$, we construct a bijection between permutations that avoid the patterns $d\cdots 1…

We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's…

We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise…

A simple convex polytope $P$ is \emph{cohomologically rigid} if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over $P$. Not every $P$ has this property, but some important polytopes such as…

This study presents a novel algorithm for identifying the set of extreme points that constitute the exact convex hull of a point set in high-dimensional Euclidean space. The proposed method iteratively solves a sequence of dynamically…

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…

Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed by the Simplex method inside the oriented graphs of polyhedra (oriented by the objective…

Let n,d be positive integers, with d even (say d=2e). Let X_(n,d) denote the locus of degree d hypersurfaces in P^n which consist of two e-fold hyperplanes. We bound the regularity of the ideal of this variety. Moreover, we show that this…

For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the…

We study the properties of the set of marginal distributions of infinite translation-invariant systems in the 2D square lattice. In cases where the local variables can only take a small number $d$ of possible values, we completely solve the…

Let $X$ be the constrained random walk on $\mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $\lambda$, $\mu_1$, $\mu_2$,...,$\mu_d$, where $\{e_1,e_2,..,e_d\}$ are the standard…

We present a simple sublinear time algorithm for testing the following geometric property. Let $P_1, ..., P_n$ be $n$ convex sets in $\mathbb{R}^d$ ($n \gg d$), such as polytopes, balls, etc. We assume that the complexity of each set…

Given a set C in R^d, let p(C) be the probability that a random d-dimensional unimodular lattice, chosen according to Haar measure on SL(d,Z)\SL(d,R), is disjoint from C\{0}. For special convex sets C we prove bounds on p(C) which are sharp…

Let $f_i(P)$ denote the number of $i$-dimensional faces of a convex polytope $P$. Furthermore, let $S(n,d)$ and $C(n,d)$ denote, respectively, the stacked and the cyclic $d$-dimensional polytopes on $n$ vertices. Our main result is that for…

We prove that for $n>3$ each generic simple polytope in $\mathbb{R}^n$ contains a point with at least $2n+4$ emanating normals to the boundary. This result is a piecewise-linear counterpart of a long-standing problem about normals to smooth…