Polytopes which are orthogonal projections of regular simplexes

We consider the polytopes which are certain orthogonal projections of k-dimensional regular simplexes in k-dimensional Euclidean space R^k. We call such polytopes -polytopes. Every sufficiently symmetric polytope, such as a regular polytope, a quasi-regular polyhedron, etc., belongs to this class. We denote by P_m,n all n-dimensional -polytopes with m vertices. We show that there is a one-to-one correspondence between the elements of P_m,n and those of P_m,m–n–1 and that this correspondence preserves the symmetry of -polytopes. Using this duality, we determine some of the P_m,n's. We also show that a -polytope is an orthogonal projection of a cross polytope if and only if it has central symmetry.