Kalai's squeezed three-spheres are polytopal
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In 1988, Kalai  extended a construction of Billera and Lee to produce many triangulated(d−1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack [2,3], he derived that for every dimension d ≥ 5, most of these(d−1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. We also give a shorter proof for Hebble and Lee’s result  that the dual graphs of these 4-polytopes are Hamiltonian.
CitationPfeifle, J. Kalai's squeezed three-spheres are polytopal. "Electronic notes in discrete mathematics", 2001, vol. 10, p. 238-241.