Blichfeldt-type inequalities and central symmetry

Abstract

Abstract A classical result of Blichfeldt, from 1921, gives a sharp lower bound on the volume of a convex body K, whose lattice points span the whole space, in terms of the lattice point enumerator . We are interested in a version of this inequality on the set of 0-symmetric convex bodies. Our motivation to study this problem comes from a lack of methods that exploit the symmetry assumption in problems of a similar kind and where 0-symmetry is a natural condition. We report upon sharp Blichfeldt-type inequalities for 0-symmetric lattice polygons, lattice crosspolytopes and lattice zonotopes.