  We identify a combinatorial quantity (the alternating sum of the h-vector)
defined for any simple polytope as the signature of a toric variety. This
quantity was introduced by Charney and Davis in their work, which in particular
showed that its non-negativity is closely related to a conjecture of Hopf on
the Euler characteristic of a non-positively curved manifold.
  We prove positive (or non-negative) lower bounds for this quantity under
geometric hypotheses on the polytope. These hypotheses lead to ampleness (or
weaker conditions) for certain line bundles on toric divisors, and then the
lower bounds follow from calculations using the Hirzebruch Signature Formula.
  Moreoever, we show that under these hypotheses on the polytope, the i-th
L-class of the corresponding toric variety is (-1)^i times an effective class
for any i.
