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 Rk. We call such polytopes π-polytopes. Every sufficiently symmetric polytope, such as a regular polytope, a quasi-regular polyhedron, etc., belongs to this class. We denote by Pm,n all n-dimensional π-polytopes with m vertices. We show that there is a one-to-one correspondence between the elements of Pm,n and those of Pm,m−n−1 and that this correspondence preserves the symmetry of π-polytopes. Using this duality, we determine some of the Pm,n's. We also show that a π-polytope is an orthogonal projection of a cross polytope if and only if it has central symmetry.