Non-Flat Regular Polytopes and Restrictions on Chiral Polytopes

摘要

An abstract polytope is flat if every facet is incident on every vertex. In this paper, we?prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures.?We then determine the three smallest non-flat regular polytopes in each rank, and?use this to show that for $n geq 8$, a chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.