A -polytope is called simple if each vertex is contained in exactly facets. A -polytope is called simplicial if each facet contains exactly vertices. By definition, a dual of simple (simlicial) polytope is simplicial (simple, respectively). Every facet of a simplicial -polytope is a -simplex. Each vertex of a simple -polytope is contained in exactly -edges.
A -cube is a simple polytope and a -simplex is both simple and simplicial.