Add time:09/05/2019         Source:sciencedirect.com

We completely classify non-spanning 3-polytopes, by which we mean lattice 3-polytopes whose lattice points do not affinely span the lattice. We show that, except for six small polytopes (all having between five and eight lattice points), every non-spanning 3-polytope P has the following simple description: P∩Z3 consists of either (1) two lattice segments lying in parallel and consecutive lattice planes or (2) a lattice segment together with three or four extra lattice points placed in a very specific manner.From this description we conclude that all the empty tetrahedra in a non-spanning 3-polytope P have the same volume and they form a triangulation of P, and we compute the h⁎-vectors of all non-spanning 3-polytopes.We also show that all spanning 3-polytopes contain a unimodular tetrahedron, except for two particular 3-polytopes with five lattice points.

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