On Primitive Subdivisions of an Elementary Tetrahedron

A polytope P of 3-space, which meets a given lattice L only in its vertices, is called L-elementary. An L-elementary tetrahedron has volume > (1/6). det(L), in case equality holds it is called L-primitive. A result of Knudsen, Mumford and Waterman, tells us that any convex polytope P admits a linear simplicial subdivision into tetrahedra which are primitive with respect to the bigger lattice (1/2) t .L, for some t depending on P. Improving this, we show that in fact the lattice (1/4).L always suffices. To this end, we first characterize all L-elementary tetrahedra for which even the intermediate lattice (1/2).L suffices.