Local optimality of the critical lattice sphere-packing of regular tetrahedra

AbstractThe critical lattice of the euclidean 3-dimensional space generated by the vertices of a regular tetrahedron T1 of side length 1 provides a 3-solid sphere-packing of any regular tetrahedron Tm derived by dilatation in the ratio m:1 from the centroid of T1. By means of a (permutation group) orbit method it is proved that that packing is locally optimal for differentiable perturbations. By means of an algebraic geometric method it is proved that the packing even is locally optimal for any small perturbation