Integration on a convex polytope

We present an exact formula for integrating a (positively) homogeneous function $f$ on a convex polytope $\Omega\subset R^n$. We show that it suffices to integrate the function on the $(n-1)$-dimensional faces of $\Omega$, thus reducing the computational burden. Further properties are derived when $f$ has continuous higher order derivatives. This result can be used to integrate a continuous function after approximation via a polynomial.