An Intersection Product for the Polytope Algebra

TL;DR

The paper introduces a novel intersection product on the polytope algebra, defined by averaging intersections $[P]\cap(x+Q)$, which endows $\Pi^*(V)$ with a unital graded commutative algebra satisfying Poincaré duality. It develops a comprehensive framework including pullbacks along linear maps, an exterior product, and a class of special elements that admit explicit linear-algebraic expressions, connecting convex geometry with valuation theory. A key set of results includes an Alexandrov–Fenchel-type inequality for higher-rank mixed volumes, a study of finite-dimensional subalgebras linked to the Dowling–Wilson conjecture, and partial validation of the degree-one hard Lefschetz and Hodge–Riemann relations, with complete proofs in dimension two and for k=1. The work bridges polytope geometry and Hodge-theoretic methods, offering a path toward resolving Dowling–Wilson-type conjectures via Lefschetz-type properties in the polytope-algebra setting.

Abstract

We introduce a new multiplication for the polytope algebra, defined via the intersection of polytopes. After establishing the foundational properties of this intersection product, we investigate finite-dimensional subalgebras that arise naturally from this construction. These subalgebras can be regarded as volumetric analogues of the graded Möbius algebra, which appears in the context of the Dowling-Wilson conjecture. We conjecture that they also satisfy the injective hard Lefschetz property and the Hodge-Riemann relations, and we prove these in degree one.

Theorems & Definitions (103)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Definition 1.5
• Theorem 1.6
• Definition 1.7
• Conjecture 1.8
• Remark 1.9
• Theorem 1.10