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...

Some gyrocommutative gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the center of momentum of three massive ob...

In this paper, we present the counterpart of Beckman–Quarles theorem in Poincaré disc model hyperbolic geometry to characterize gyroisometries (hyperbolic isometries) with a single nonzero distance a ∈ 0,1 satisfying id="M2"> <...