# Non-flat regular polytopes and restrictions on chiral polytopes

## TL;DR

This paper investigates restrictions on chiral polytopes with flat facets, proving they cannot have flat regular facets and vertex-figures, and determines minimal non-flat regular polytopes, establishing lower bounds on flags for higher ranks.

## Contribution

It proves that chiral polytopes cannot have flat regular facets and vertex-figures, and identifies the smallest non-flat regular polytopes in each rank, providing new bounds on flags.

## Key findings

- No chiral polytope has flat finite regular facets and vertex-figures.
- Determined the three smallest non-flat regular polytopes in each rank.
- For n ≥ 8, a chiral n-polytope has at least 48(n-2)(n-2)! flags.

## Abstract

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 $n \geq 8$, a chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.