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

Given a convex body K ⊂ R ^d, what is the probability that a randomly chosen congruent copy, K^*, of K is lattice‐point free, that is, K^*∩ Z ^d = ∅︁? 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. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007