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Are there any purely combinatorial criteria which allow you to deduce that a spherical simplicial complex is polytopal (i.e., there exists a simplicial polytope whose boundary is isomorphic to it)?

For example, I was kind of hoping that shellability might be enough, but this is not true. All 3-spheres on up to ten vertices are shellable, but not all are polytopal, as I learned from this paper by Frank Lutz.

This question is not the same as the question What is the best way to test if a sphere is a polytope since that question (at least as it was answered) considers how to carry out the test on any input, while I am interested in a sufficient criterion (which might work only rarely, but hopefully on nice examples).

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I can think of a few purely combinatorial criteria, that allow to deduce realizability as a polytope.

  1. All d-polytope with at most d+2 vertices is realizable

  2. Stacked polytopes. (It can be easily combinatorially checked if a simplicial complex is stacked.)

Those and some more obscure criteria a mentioned for example in a paper by Stefan Felsner and Sarah Kappes: https://arxiv.org/abs/math/0602063v1

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