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 Γ, either as kernels of Γ* or as images of Γ. Entries of Γ 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 Γ determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c=1.