On convex bodies that permit packings of high density

Abstract

It is well known that an n-dimensional convex body permits a lattice packing of density 1 only if it is a centrally symmetric polytope of at most 2(2n-1) facets. This article concerns itself with the associated stability problem whether a convex body that permits a packing of high density is in some sense close to such a polytope. Several inequalities that address this stability problem are proved. A corresponding theorem for coverings by two-dimensional convex bodies is also proved. © 1990 Springer-Verlag New York Inc.