Minimal equivelar polytopes

Abstract

Every equivelar abstract polytope of type { p 1 , …, p n − 1 } has at least 2 p 1 ⋯ p n − 1 flags. In this paper, we study polytopes that attain this lower bound, called tight polytopes . Using properties of flat polytopes, we are able to give a complete local characterization of when a polytope is tight. We then show a way to construct tight polyhedra of type { p , q } when p and q are not both odd, and a way to construct regular tight polytopes of type {2 k 1 , …, 2 k n − 1 } .