A generalization of Filliman duality

Filliman duality expresses (the characteristic measure of) a convex polytope $P$ containing the origin as an alternating sum of simplices that share supporting hyperplanes with $P$. The terms in the alternating sum are given by a triangulation of the polar body $P^{\circ}$. The duality can lead to useful formulas for the volume of $P$. A limiting case called Lawrence's algorithm can be used to compute the Fourier transform of $P$.

In this note we extend Filliman duality to an involution on the space of polytopal measures on a finite-dimensional vector space, excluding polytopes that have a supporting hyperplane coplanar with the origin. As a special case, if $P$ is a convex polytope containing the origin, any realization of $P^{\circ}$ as a linear combination of simplices leads to a dual realization of $P$.