On the Order Dimension of Convex Polytopes

Given a convex polytope P, how does the usual affine dimension dimp(P) correlate with dimO(L(P)), the order dimension of the face lattice of P? It will be shown that dimP(P) + 1 ⩽ dimO(P)), that equality holds for some of the standard examples, but that the order dimension of cyclic polytopes C(n, d) can be arbitrarily large for fixed d ⩾ 4. Using the Four Colour Theorem it is easy to show that a simple 3-dimensional convex polytope has order dimension at most 8