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](https://arxiv.org/pdf/math/0604018.pdf) 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](https://mathoverflow.net/questions/50100/whats-the-best-way-to-test-if-a-sphere-is-a-polytope-algorithms-for-the-simpl) 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).



Sufficient criterion for a simplicial sphere to be polytopal
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> 


