Non-flat regular polytopes and restrictions on chiral polytopes

An abstract polytope is \emph{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 $n \geq 8$,achiral$n$-polytopehasatleast$48(n-2)(n-2)!$ flags.