Universidad de Cantabria
Thursday, April 24 at 3:00 PM
A lattice d-simplex is the convex hull of d+1 affinely independent integer points in Rd. It is called empty if it contains no lattice point apart of its d + 1 vertices. The classification of empty 3-simplices is known since 1964 (White), based on the fact that they all have width one. But for dimension 4 no complete classification is known.
Haase and Ziegler (2000) computed all empty 4-simplices up to determinant 1000 and based on their results conjectured that after determinant 179 all empty 4-simplices have width one or two. We prove this conjecture as follows:
– We show that no empty 4-simplex of width three or more can have determinant greater than 7000, by combining the recent classification of hollow 3-polytopes (Averkov, Krümpelmann and Weltge) with general methods from the geometry of numbers.
– We continue the computations of Haase and Ziegler up to determinant 7600, and find that no new 4-simplices of width larger than two arise. In particular, we give the whole list of empty 4-simplices of width larger than two, which is as computed by Haase and Ziegler.
In the case of empty 4-simplices of width not greater than 2 we present some new families that were not classified before. This results will lead to a complete classification of empty 4-simplices with the help of some additional upper bounds for the volume of this particular simplices.
Joint work with Francisco Santos.
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