Motivated by problems of hyperbolic stochastic geometry we introduce and study the class beta-star polytopes. A polytope is defined as convex hull an inhomogeneous Poisson processes on complement unit ball in Rd with density proportional to (‖x‖2−1)−β, where ‖x‖>1 β>d/2. Explicit formulas for various geometric combinatorial functionals associated polytopes are provided, including expected numbe...

A binary space partitioning (bsp) tree represents a recursive, hierarchical subdivision of d-dimensional space into convex subspaces. BSP trees provide a computational representation of space that provides both a search structure and a representation of geometry. Constructing a bsp tree consists of partitioning a subspace by means of a hyperplane, which intersects the interior of that subspace,...

Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon’s descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types A,B,C, H3, and rank two groups. Both notions have the same, simple eigenvalues. The hyperplane definition is especially natural and satisfies a positivity property when W is crystallographic and the...

The anchored hyperplane location problem is to locate a hyperplane passing through some given points P ⊆ IR and minimizing either the sum of weighted distances (median problem), or the maximum weighted distance (center problem) to some other points Q ⊆ IR. This problem of computational geometry is analyzed by using nonlinear programming techniques. If the distances are measured by a norm, it wi...

R. Thomas (with a remark of B. Totaro) proved that the Hodge conjecture is essentially equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomology of a general fiber, and show that each relation between t...

Abstract Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon’s descent algebra and another using random walk on chambers of hyperplane arrangements. These definitions coincide for types A,B,H3, and rank two groups. Both notions satisfy a convolution property and have the same simple eigenvalues. The hyperplane definition is especially natural and satisfies a po...