LATTICE COVERINGS AND GAUSSIAN MEASURES OF N-DIMENSIONAL CONVEX-BODIES

Let parallel to . parallel to be the euclidean norm on R(n) and let ga mma(n), be the (standard) Gaussian measure on R(n) with density (2 pi) (-n/2)e(-parallel to x parallel to 2/2). Let curly theta (similar or e qual to 1.3489795) be defined by gamma(1)([-curly theta/2, curly theta /2]) = 1/2 and let L be a lattice in R(n) generated by vectors of norm less than or equal to curly theta. Then, for any closed convex set V in R(n) with gamma(n)(V) greater than or equal to 1/2, we have L + V = R(n) (equivalently, for any a epsilon R(n), (a + L) boolean AND V not equal 0). The above statement can also be viewed as a ''nonsymmetric' ' version of the Minkowski theorem.