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

Abstract

Given a convex body K ⊂Rd, what is the probability that a randomly chosen congruent copy, K*, of K is lattice-point free, that is, K*∩Zd = ∅︁? Here Zd is the usual lattice of integer points in Rd. 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