An upper bound for the diameter of a polytope

The distance between two vertices of a polytope is the minimum number of edges in a path joining them. The diameter of a polytope is the greatest distance between two vertices of the polytope. We show that if P is a d-dimensional polytope with n facets, then the diameter of P is at most $\text{1}\text{3}\text{2}{}^{\mathrm{d?3}}\text{(n?d+}\text{5}\text{2}\text{)}$.