Tight orientably-regular polytopes

Abstract

It is known that every equivelar abstract polytope of type {p₁, …, p_n − 1} has at least 2p₁⋯p_n − 1 flags. Polytopes that attain this lower bound are called tight. Here we investigate the conditions under which there is a tight orientably-regular polytope of type {p₁, …, 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 (when defined) are even divisors of 2p_i.