On convex bodies that permit packings of high density (Q921595)

A theorem of Minkowski asserts that an n-dimensional convex body permits a lattice packing of density 1 only if it is a centrally symmetric polytope with at most \(2(2^ n-1)\) facets. The author addresses the associated stability problem: whether a convex body that permits a packing of high density is in some sense close to such a polytope. If K is a convex body in \(E^ n\) and \(S\subset E^ n\), the collection \({\mathcal P}(K,S)=\{K+s:\) \(s\in S\}\) is called a translative packing in \(E^ n\) if any two members of \({\mathcal P}(K,S)\) have no interior points in common. Let h denote Hausdorff distance and let d denote diameter. The author proves the following theorem: Let K be a convex body in \(E^ n\) that permits a translative packing of density \(\delta\). Then there is a centrally symmetric convex body \(K^ 0\) (which is the translate of \(K^*\) whose center coincides with the centroid of K) such that \[ h(K,K^ 0)\leq \theta_ nd(K)((1-\delta)/\delta)^{1/(n+1)}, \] where \(\theta_ n=4\cdot 2^{1/(n+1)}\cdot 3^{(n-1)/n} n^{n/(n+1)}<12n\). Furthermore, there exist an affine copy \(\tilde K\) of K and a normalized convex body Z such that Z is centrally symmetric with respect to the centroid of \(\tilde K\) and \[ h(\tilde K,Z)\leq \mu_ n(1+\delta)^{1/(n+1)}, \] where \(\mu_ n=(8/\kappa_ n^{1/n})4^{1/(n+1)}\cdot 3^{(n-1)/n} n^{(3n+1)/2(n+1)}<12n^ 2.\) He also proves an analogous theorem concerning the Hausdorff distance between i) centrally symmetric convex bodies in \(E^ n\) that permit lattice packings of high density \(<1\) and polytopes of the kind discussed above and ii) centrally symmetric convex bodies in \(E^ 2\) that permit translative coverings of density \(\delta <1\) and certain hexagons.