Beta-star polytopes and hyperbolic stochastic geometry

Stochastic Geometry: Random Polytopes

Since today’s theme is geometry and probability, we start with a very cute problem combining both. Recall that a set S ∈ R is called convex if between every two points x, y the segment [x, y] lies inside S. If S ∈ R is an arbitrary set then its convex hull, conv(S) is the minimal convex set containing all of S. Problem: If we choose n points uniformly in the unit circle, what is the probability...

On hyperbolic virtual polytopes and hyperbolic fans

Abstract: Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ R3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R1 ≤ C ≤ R2 then K is a ball. (R1 and R2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous constructio...

Essential Hyperbolic Coxeter Polytopes

We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimens...

Pointed Spherical Tilings and Hyperbolic Virtual Polytopes

The paper presents an introduction to the theory of hyperbolic virtual polytopes from the viewpoint of combinatorial rigidity theory. Namely, we give a shortcut for a reader who is acquainted with the notions of Laman graphs, 3D liftings and pointed tilings. From this viewpoint, a hyperbolic virtual polytope is a stressed pointed graph embedded in the sphere S. The advantage of such a presentat...

Hyperbolic Geometry