Lattice coverings and Gaussian measures of n-dimensional convex bodies

Abstract

Let ∥ · ∥ be the euclidean norm on Rn and let γn be the (standard) Gaussian measure on Rn with density (2π)-n/2e-∥x∥2/2. Let θ (≃ 1.3489795) be defined by γ1([-θ/2, θ/2]) = 1/2 and let L be a lattice in Rn generated by vectors of norm ≤ θ. Then, for any closed convex set V in Rn with γn(V) ≥ 1/2, we have L + V = Rn (equivalently, for any a ∈ Rn, (a + L) ∩ V ≠ 0∅). The above statement can also be viewed as a "nonsymmetric" version of the Minkowski theorem.