  We introduce a parameter space for periodic point sets, given as unions of
$m$ translates of point lattices. 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 $d=24$.
