The chance that a convex body is lattice-point free:: A relative of Buffon's needle problem

Given a convex body K subset of R(d), what is the probability that a randomly chosen congruent copy, K*, of K is lattice-point free, that is, K* boolean AND Z(d) = circle divide? Here Z(d) is the usual lattice of integer points in R(d). Luckily, the underlying probability is well defined since integer translations of K can be factored out. The question came up in connection with integer programming. We explain what the answer is for convex bodies of large enough volume. (c) 2006 Wiley Periodicals, Inc.