Let K be a convex body in R. A random polytope is the convex hull [x1, ..., xn] of finitely many points chosen at random in K. E(K, n) is the expectation of the volume of a random polytope of n… (More)

“Almost exact” estimates for the Delone triangulation numbers are given. In particular lim n→∞ deln−1 n = 1 2πe = 0.0585498.... 1991 Mathematics Subject Classification. 52A22. The first named author… (More)

Let Xi,j , i, j = 1, ..., n, be independent, not necessarily identically distributed random variables with finite first moments. We show that the norm of the random matrix (Xi,j) n i,j=1 is up to a… (More)

For a given sequence of real numbers a1, . . . , an we denote the k-th smallest one by kmin1≤i≤n ai. Let A be a class of random variables satisfying certain distribution conditions (the class… (More)

In contemporary convex geometry, the rapidly developing Lp-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the Lp-affine surface… (More)

There exist positive constants c0 and c1 = c1(n) such that for every 0 < < 1/2 the following holds: Let P be a convex polytope in R having N ≥ c0/ vertices x1, . . . , xN . Then there exists a subset… (More)

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ1, . . . , ξn with log-concave distribution and scalars x1, . . . , xn,… (More)

We study geometric parameters associated with the Banach spaces (IRn, ‖·‖k,q) normed by ‖x‖k,q = (∑ 1≤i≤k |〈x, ai〉| )1/q , where {ai}i≤N is a given sequence of N points in IRn, 1 ≤ k ≤ N , 1 ≤ q ≤ ∞… (More)

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far… (More)