Diameters of sections and coverings of convex bodies

We study the diameters of sections of convex bodies in RN determined by a random N × n matrix , either as kernels of ∗ or as images of . Entries of 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 RN 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 determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c = 1. © 2005 Elsevier Inc. All rights reserved