Type: Article

The probability that a convex body intersects the integer lattice in a k-dimensional set

Roldán-Pensado, Edgardo;

Abstract:

Let K be a convex body in the Euclidean d-dimensional space. It is known that there is a constant C0 depending only on d such that the probability that a random copy ?(K) of K does not intersect the lattice Zd is smaller than C0/|K| and this is sharp. In this paper the author proves that for every k<d there is a constant C such that the probability that ?(K) contains a subset of dimension k is smaller than C/|K| (this is the best possible if k=d?1).