The Order Dimension of Convex Polytopes

With a convex polytope ${\text{M}}$ in $\mathbb{R}^3$, a partially ordered set ${\text{P}}_{\text{M}} $ is associated whose elements are the vertices, edges, and faces of ${\text{M}}$ ordered by inclusion. This paper shows that the order dimension of ${\text{P}}_{\text{M}} $ is exactly 4 for every convex polytope ${\text{M}}$. In fact, the subposet of ${\text{P}}_{\text{M}} $ determined by the vertices and faces is critical in the sense that deleting any element leaves a poset of dimension 3.