Non-flat regular polytopes and restrictions on chiral polytopes (Q2409823)

Summary: An abstract polytope is \textit{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.