On Empty Simplices, Pyramids, Parallelepipeds

Abstract

AbstractAn integer polyhedron is called empty if it does not contain integer points other than its vertices. In this chapter we give the classification of empty tetrahedra and the classification of pyramids whose integer points are contained in the base of pyramids in \(\mathbb{R}^{3}\). Later in the book we essentially use the classification of the mentioned pyramids for studying faces of multidimensional continued fractions. In particular, the describing of such pyramids simplifies the deductive algorithm of Chap. 20 in the three-dimensional case. We continue with two open problems related to empty objects in lattices. The first one is a problem of description of empty simplices in dimensions greater than 3. The second is the lonely runner conjecture. We conclude this chapter with a proof of a theorem on the classification of empty tetrahedra.KeywordsInteger PointInteger LatticeLeft CosetUniversal Covering SpaceEmpty TriangleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.