Recent Results on Random Polytopes

Geometry of random (-1/+1)-polytopes A survey of recent results on random {−1, 1}-polytopes in Asymptotic Convex Analysis

Random {−1, 1}-polytopes demonstrate extremal behavior with respect to many geometric characteristics.

Recent Results on Random Polytopes

Linear Programming, the Simplex Algorithm and Simple Polytopes

In the first part of the paper we survey some far reaching applications of the basis facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concurring the simplex algorithm. We describe sub-exponential randomized pivot roles and upper bounds on the diameter of graphs of polytopes. 

On the variance of random polytopes

A random polytope is the convex hull of uniformly distributed random points in a convex body K. A general lower bound on the variance of the volume and f -vector of random polytopes is proved. Also an upper bound in the case when K is a polytope is given. For polytopes, as for smooth convex bodies, the upper and lower bounds are of the same order of magnitude. The results imply a law of large n...

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...