The University of Michigan Combinatorics Seminar


Abstract 

The notion of a two dimensional angle is naturally extended to an
ndimensional
'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 ndimensional theta function over an
ndimensional
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.
