Polytope volume by descent in the face lattice and applications in social choice

TL;DR

This paper introduces a descent-based algorithm for computing polytope volumes, implemented in Normaliz, with applications in social choice theory where volumes relate to voting paradoxes.

Contribution

It presents a novel descent in face lattice method for volume computation, connecting it to triangulations and demonstrating efficiency on high-dimensional polytopes.

Findings

Efficient volume computation demonstrated on high-dimensional polytopes

Connection established between face lattice descent and reverse-lexicographic triangulations

Application to voting theory probabilities involving polytope volumes

Abstract

We describe the computation of polytope volumes by descent in the face lattice, its implementation in Normaliz, and the connection to reverse-lexicographic triangulations. The efficiency of the algorithm is demonstrated by several high dimensional polytopes of different characteristics. Finally, we present an application to voting theory where polytope volumes appear as probabilities of certain paradoxa.