An Intersection Product for the Polytope Algebra

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 M\"obius 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.