Dual Mixed Volume

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 $φ^3$-scalar amplitude at tree level.

Figures (7)

• Figure 1: A polytope $P$ and its normal fan $\mathcal{N}(P)$
• Figure 2: An unbounded polyhedron $P$ and its normal fan $\mathcal{N}(P)$
• Figure 3: A fine mixed subdivision of the Minkowski sum of two triangles
• Figure 4: Left: $P$ and $[-{\mathbf{p}},{\mathbf{p}}]$. Right: $\mathcal{N}(P)$ (undashed) and $\mathcal{N}(P+[-{\mathbf{p}},{\mathbf{p}}])$.
• Figure 5: A spanning tree $G_{{\mathcal{J}}}$

Theorems & Definitions (146)

• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Lemma 2.6
• Definition 2.7
• Proposition 2.8