Tight orientably-regular polytopes

Abstract

Every equivelar abstract polytope of type $\{p_1, \ldots, p_{n-1}\}$ has at least $2p_1 \cdots p_{n-1}$ flags. Polytopes that attain this lower bound are called tight. Here we investigate the question of under what conditions there is a tight orientably-regular polytope of type $\{p_1, \ldots, p_{n-1}\}$. We show that it is necessary and sufficient that whenever $p_i$ is odd, both $p_{i-1}$ and $p_{i+1}$ are even divisors of $2p_i$.