An indexed set of density bounds on lattice packings (Q1345118)

For the lattice packing density \(\delta_ L\) of an \(O\)-symmetric convex body \(B\) (i.e. compact and with \(\text{int } B\neq \emptyset\)) in Euclidean \(n\)-space, \(n\geq 2\) the author proves the following theorem: Suppose \(j\) is an integer with \(0\leq j\leq n/2\) and \(p\) an odd prime with \(p^ j\geq \text{card} (\Gamma \cap \text{int } B)-1\), where \(p\Gamma\) is a lattice of determinant \(\neq 0\) and admissible for \(B\). Then holds for the lattice packing density \[ \delta_ L (B)\geq \text{Vol} (B) (2^ n p^ j \text{ det } \Gamma)^{-1}. \] This generalizes earlier results for \(j=0\) and \(j=1\).