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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.
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Thomas Wannerer (2026) studied this question.
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