Variations of Minkowski's theorem on successive minima

Abstract

Abstract Minkowski's second theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of an o -symmetric convex body in terms of its successive minima. In this paper we study analogs of this theorem from two different points of view: either relaxing the symmetry condition, assuming that the centroid of the body lies at the origin, or replacing the volume functional by the surface area.