网页Theorem.(Minkowski's theorem) Suppose C to be a symmetric, convex, bounded subset Rd. If vol( C ) > 2d, then C contains at least one lattice point other than 0. Proof. Take C0= 1 2 C = 1 2 x : x 2 C , then vol( C0) > 1. Then we claim that There exists a nonzero integral vector v 2 Zdnf 0g, such that v 2 C0C0.