Recently Bey, Henk, and Wills proved that for a polytope [special characters omitted] with k interior lattice points, the volume of P satisfies[special characters omitted]The main focus of this dissertation is to show via triangulations that for d ≥ 3 there is exactly one class (under unimodular equivalence) of nondegenerate lattice simplices in [special characters omitted] with k ≥ 1 interior lattice points and volume [special characters omitted].
We begin by showing that minimal volume occurs if and only if the P is a lattice simplex (of dimension d ≠ 2) whose interior lattice points are collinear with a vertex of P. We then show that there can only be one such class of simplices with this property. Interestingly, this statement is not true for d = 2, and counterexamples are provided within.
|School:||University of Illinois at Urbana-Champaign|
|School Location:||United States -- Illinois|
|Source:||DAI-B 69/11, Dissertation Abstracts International|
|Keywords:||Lattice points, Simplices, Triangulations, Unimodular transformations|
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