There have 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 cases of 4-dimensional polytopes and 3-dimensional spheres. One crucial question is the “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. (Joint work with Philip Brinkmann)

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