Covering minima and lattice point free convex bodies

Abstract

Suppose K is a convex set of nonzero volume in Euclidean n-space R ⁿ and it is symmetric about the origin (i.e., if x belongs to K, so does - x). For any real number t, let tK={tx : x∈K}. The infimum over all positive real numbers t such that if a copy of tK is placed centered at every integer point, all of R ⁿ is covered, is called the “covering radius” of K (with respect to the lattice Z ⁿ). The covering radius and related quantities have been studied extensively in Geometry of Numbers. In this paper, we define and study the “covering minima” of a convex body which is not necessarily symmetric about the origin; the covering radius will be a special case of one of of these minima. This extension to general convex bodies has among other things, applications to algorithms for Integer Programming which was our initial motivation. This motivation is explained in some detail later. We use the results of the paper to derive bounds on the width of lattice point free convex bodies and analyze their structure.

Department of Computer Science, Carnegie-Mellon University, Pittsburgh; supported by NSF Grant ECS-8418392