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.

Este artículo explora los viajes en el tiempo y sus implicaciones.

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