Focus Area 2: Delaunay Geometry
A seminal result in the theory of toric varieties, due to Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope P there is a positive integer k such that the dilated polytope kP has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that k=4 works for every polytope. But this does not imply that every k>4 works as well. We here study the values of k for which the result holds showing that:
1. It contains all composite numbers.
2. It is an additive semigroup.
These two properties imply that the only values of k that may not work (besides 1 and 2, which are known not to work) are k∈{3,5,7,11}. With an ad-hoc construction we show that k=7 and k=11 also work, except in this case the triangulation cannot be guaranteed to be "standard" in the boundary. All in all, the only open cases are k=3 and k=5.