by Achill Schürmann.

**Series:** 2008-20, Preprints

- MSC:
- 11H55 Quadratic forms (reduction theory, extreme forms, etc.)
- 52C17 Packing and covering in $n$ dimensions

**Abstract:**

We introduce a parameter space for periodic point sets, given as a union of $m$~translates of a point lattice.

In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect,

strongly eutactic lattices cannot be locally

improved to yield a periodic sphere packing

with greater density. This applies in particular

to the densest known lattice sphere packings

in dimension $d\leq 8$ and $d=24$.

**Keywords:**

sphere packing, root lattice, Leech lattice, periodic point set, positive definite quadratic form, Voronoi's characterization, perfect lattice, strongly eutactic lattice