Projected Products of Polygons

in conjunction with Euler’s formula, f1 = f0 + f2 − 2; see also [6, Sect. 10.3]. Moreover, the cone of flag vectors of 3-polytopes is also given by these two inequalities, together with the usual Dehn– Sommerville relations, such as f02 = 2f1 and f012 = 4f1. Here the extreme cases, polytopes whose f or flag vectors lie at the boundary of the cone, are given by the simplicial polytopes (for which Steinitz’ first inequality is tight) and the simple polytopes (second inequality tight). Moreover, all the integer points that satisfy the above conditions do correspond to actual polytopes. For 4-dimensional polytopes, such a complete and simple answer is not to be expected; compare Grünbaum [6, Sect. 10.4], Bayer [3] and Höppner & Ziegler [7]. However, we may now be getting close to a full description of the convex cone spanned by the f -vectors (with apex at the f -vector of the 4-simplex): In the projective coordinates introduced in [17],