Polytopes which are orthogonal projections of regular simplexes (Q756137)

The author discusses the class \(P_{m,n}\) of convex n-polytopes in \({\mathbb{R}}^ n\) with m vertices which are certain orthogonal projections of regular simplices. The members in \(P_{m,n}\) are called \(\pi\)- polytopes. Every sufficiently symmetric polytope is an example of a \(\pi\)-polytope. The author proves that there exists a one-to-one correspondence (``duality'') between the polytopes in \(P_{m,n}\) and those in \(P_{m,m-n-1}\) which preserves the symmetry groups. Also it is shown that a \(\pi\)-polytope is an orthogonal projection of a crosspolytope if and only if it has central symmetry.