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.