  Given a set C in R^d, let p(C) be the probability that a random d-dimensional
unimodular lattice, chosen according to Haar measure on SL(d,Z)\SL(d,R), is
disjoint from C\{0}. For special convex sets C we prove bounds on p(C) which
are sharp up to a scaling of C by a constant. We also prove bounds on a variant
of p(C) where the probability is conditioned on the random lattice containing a
fixed given point p. Our bounds have applications, among other things, to the
asymptotic properties of the collision kernel of the periodic Lorentz gas in
the Boltzmann-Grad limit, in arbitrary dimension d.
