  A simple convex lattice polytope $\Box$ defines a torus-equivariant line
bundle $\LB$ over a toric variety $\XB.$ Atiyah and Bott's Lefschetz
fixed-point theorem is applied to the torus action on the $d''$-complex of
$\LB$ and information is obtained about the lattice points of $\Box$. In
particular an explicit formula is derived, computing the number of lattice
points and the volume of $\Box$ in terms of geometric data at its extreme
points. We show this to be equivalent the results of Brion \cite{brion} and
give an elementary convex geometric interpretation by performing Laurent
expansions similar to those of Ishida \cite{ishida}.
