  A subset K of R^n gives rise to a formal Laurent series with monomials
corresponding to lattice points in K. Under suitable hypotheses, this series
represents a rational function R(K). Michel Brion has discovered a surprising
formula relating the rational function R(P) of a lattice polytope P to the sum
of rational functions corresponding to the supporting cones subtended at the
vertices of P. The result is re-phrased and generalised in the language of
cohomology of line bundles on complete toric varieties. Brion's formula is the
special case of an ample line bundle on a projective toric variety. - The paper
also contains some general remarks on the cohomology of torus-equivariant line
bundles on complete toric varieties, valid over noetherian ground rings.

Combinatorics;Algebraic Geometry