The Probability that a Convex Body Intersects the Integer Lattice in a k-dimensional Set

Let K be a convex body in a"e (d) . It is known that there is a constant C (0) depending only on d such that the probability that a random copy rho(K) of K does not intersect a"currency sign (d) is smaller than C-0/vertical bar K vertical bar and this is best possible. We show that for every k < d there is a constant C such that the probability that rho(K) contains a subset of dimension k is smaller than C/vertical bar K vertical bar. This is best possible if k=d-1. We conjecture that this is not best possible in the rest of the cases; if d=2 and k=0 then we can obtain better bounds. For d=2, we find the best possible value of C (0) in the limit case when width(K)-> 0 and |K|-> a.