On Primitive Subdivisions of an Elementary Tetrahedron

J.-M. Kantor and K.S. Sarkaria

Vol. 211 (2003), No. 1, 123–155

DOI: 10.2140/pjm.2003.211.123

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

Milestones

Received: 24 March 2002

Revised: 23 October 2002

Published: 1 September 2003