Lattice reduced and complete convex bodies

Abstract

The purpose of this paper is to study convex bodies (Formula presented.) for which there exists no convex body (Formula presented.) of the same lattice width. Such bodies will be called ‘lattice reduced’, and they occur naturally in the study of the flatness constant in integer programming, as well as other problems related to lattice width. We show that any simplex that realizes the flatness constant must be lattice reduced and prove structural properties of general lattice reduced convex bodies: they are polytopes with at most (Formula presented.) vertices and their lattice width is attained by at least (Formula presented.) independent directions. Strongly related to lattice reduced bodies are the ‘lattice complete bodies’, which are convex bodies (Formula presented.) for which there exists no (Formula presented.) such that (Formula presented.) has the same lattice diameter as (Formula presented.). Similar structural results are obtained for lattice complete bodies. Moreover, various construction methods for lattice reduced and complete convex bodies are presented.