THE STRUCTURE OF THE POLYTOPE ALGEBRA

We construct an isomorphism from McMullen's polytope algebra, onto the quotient of the algebra of continuous, piecewise polynomial functions with integral value at 0, by its ideal generated by coordinate functi ons. This explains the non-trivial grading of the polytope algebra, by the obvious grading of piecewise polynomial functions. In the process of the proof, we make explicit many connections between convex polyto pes and piecewise polynomials.