Dual Mixed Volume

TL;DR

This paper introduces the dual mixed volume, a new rational function for polytopes, exploring its properties, relations to other polynomials, and applications to specific polytope classes and scalar amplitudes.

Contribution

It defines the dual mixed volume, analyzes its properties, and connects it to various polytopes and physical amplitudes, providing new insights and tools.

Findings

Dual mixed volume is additive under mixed subdivisions.

It relates to the dual volume of Cayley polytopes via a change of variables.

Reproduces planar $3$-scalar amplitude at tree level.

Abstract

We define and study the dual mixed volume rational function of a sequence of polytopes, a dual version of the mixed volume polynomial. This concept has direct relations to the adjoint polynomials and the canonical forms of polytopes. We show that dual mixed volume is additive under mixed subdivisions, and is related by a change of variables to the dual volume of the Cayley polytope. We study dual mixed volume of zonotopes, generalized permutohedra, and associahedra. The latter reproduces the planar $\phi^3$-scalar amplitude at tree level.