Solid angles and theta functions on polytopes

The notion of a two dimensional angle is naturally extended to an n-dimensional 'solid' angle by considering polyhedral cones. Macdonald has studied the sum of solid angles over all lattice points in a polytope. Here we use Fourier analysis to first integrate the classical n-dimensional theta function over an n-dimensional polytope P, and then allow the time parameter in the theta function to approach zero. In the limit, an interesting polynomial valuation on polytopes is obtained (involving solid angles) and explicit formulas are given for this valuation on polytopes, namely for its codimension two term. Thus solid angles provide a link between theta functions and polytopes.