  Ordinary polytopes were introduced by Bisztriczky as a (nonsimplicial)
generalization of cyclic polytopes. We show that the colex order of facets of
the ordinary polytope is a shelling order. This shelling shares many nice
properties with the shellings of simplicial polytopes. We also give a shallow
triangulation of the ordinary polytope, and show how the shelling and the
triangulation are used to compute the toric h-vector of the ordinary polytope.
As one consequence, we get that the contribution from each shelling component
to the h-vector is nonnegative. Another consequence is a combinatorial proof
that the entries of the h-vector of any ordinary polytope are simple sums of
binomial coefficients.
