A note on volume thresholds for random polytopes

Volume Thresholds for Gaussian and Spherical Random Polytopes and Their Duals

Let g be a Gaussian random vector in R. Let N = N(n) be a positive integer and let KN be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes VN := E vol(KN ∩RB 2 )/ vol(RB 2 ). For a large range of R = R(n), we establish a sharp threshold for N , above which VN → 1 as n → ∞, and below which VN → 0 as n → ∞. We also consider the case when KN is generated by ...

A Note on Palindromic Δ-vectors for Certain Rational Polytopes

Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi’s Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart δ-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact.

Recent Results on Random Polytopes

Random Polytopes on the Torus

The expected volume of the convex hull of n random points chosen independently and uniformly on the d-dimensional torus is determined. In the 1860's J. J. Sylvester raised the problem of determining the expected area V$(C) of the convex hull of three points chosen independently and uniformly at random from a given plane convex body C of area one. For some special plane convex bodies the problem...

Recent Results on Random Polytopes

This is a survey over recent asymptotic results on random polytopes in d-dimensional Euclidean space. Three ways of generating a random polytope are considered: convex hulls of finitely many random points, projections of a fixed highdimensional polytope into a random d-dimensional subspace, intersections of random closed halfspaces. The type of problems for which asymptotic results are describe...