  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, 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.
  Moreover, we can now give a shorter proof of Hebble & Lee's 2000 result that
the dual graphs of these 4-polytopes are Hamiltonian. Therefore, the polars of
these Kalai polytopes yield another family supporting Barnette's conjecture
that all simple 4-polytopes admit a Hamiltonian circuit.
