There has been massive efforts to understand the parameter spaces of convex polytopes -- and great results such as the “g-Theorem” were achieved on the way. On the other hand, key questions are still open, already and in particular for the case of 4-dimensional polytopes/3-dimensional spheres. One crucial question is “fatness problem” for 4-dimensional polytopes, which is a key to the question whether we should expect the same answers for convex polytopes (which are discrete-geometric objects) and for cellular spheres (a topological model), at least asymptotically.

I will argue that we should not, and present first results in this direction: The sets of f-vectors of 3-spheres and of 4-polytopes do not coincide. One of the efficient key tools in the computations for that is the “biquadratic final polynomials” which may be derived from a linearization of the non-linear (quadratic) Grassmann-Plücker relations. How about inscribable polytopes - to add another non-linear condition?

(Joint work with Philip Brinkmann and others)