Sharp Concentration of Random Polytopes

Smallest singular value of random matrices and geometry of random polytopes

We study behaviour of the smallest singular value of a rectangular random matrix, i.e., matrix whose entries are independent random variables satisfying some additional conditions. We prove a deviation inequality and show that such a matrix is a “good” isomorphism on its image. Then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes (absolute...

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 ...

On the average number of sharp crossings of certain Gaussian random polynomials

Banach-Mazur Distances and Projections on Random Subgaussian Polytopes

We consider polytopes in Rn that are generated by N vectors in Rn whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random ±1 polytopes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur distance between them is essentially of a maximal order n. This result is a...

Classifications and volume bounds of lattice polytopes

In this licentiate thesis we study relations among invariants of lattice polytopes, with particular focus on bounds for the volume. In the first paper we give an upper bound on the volume vol(P∗) of a polytope P∗ dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d . This bound, expressed in terms of the Sylvester sequence, is sharp, and is ach...