Diameters of sections and coverings of convex bodies

摘要

We study the diameters of sections of convex bodies in RN determined by a random N x n matrix Gamma, either as kernels of Gamma* or as images of Gamma. Entries of Gamma are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random variables. We show that if a symmetric convex body K in R-N has one well bounded k-codimensional section, then for any m > ck random sections of K of codimension m are also well bounded, where c >= 1 is an absolute constant. It is noteworthy that in the Gaussian case, when Gamma determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c = 1. (c) 2005 Elsevier Inc. All rights reserved.