Volumes of Polytopes Without Triangulations

TL;DR

This paper introduces a triangulation-independent formalism for computing polytope volumes using vertex objects, enabling new insights into the geometry of the dual amplituhedron and simplifying volume calculations.

Contribution

It presents a novel vertex-based calculus for polytope volumes that is independent of triangulations and applicable to the dual amplituhedron geometry.

Findings

Derived new volume expressions for the NMHV dual amplituhedron

Provided a triangulation-independent formalism for polytope volumes

Facilitated the derivation of identities among different triangulations

Abstract

The geometry of the dual amplituhedron is generally described in reference to a particular triangulation. A given triangulation manifests only certain aspects of the underlying space while obscuring others, therefore understanding this geometry without reference to a particular triangulation is desirable. In this note we introduce a new formalism for computing the volumes of general polytopes in any dimension. We define new "vertex objects" and introduce a calculus for expressing volumes of polytopes in terms of them. These expressions are unique, independent of any triangulation, manifestly depend only on the vertices of the underlying polytope, and can be used to easily derive identities amongst different triangulations. As one application of this formalism, we obtain new expressions for the volume of the tree-level, $n$-point NMHV dual amplituhedron.