Abstract

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 sufices. To this end, we first characterize all L-elementary tetrahedra for which even the intermediate lattice (1 ∕ 2).L sufices.

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