Abstract
This paper explores the metrical properties of convex polytopes by means of the classical Plücker embedding of the GrassmannianG(k, n) ofk-planes inR n into the exterior algebra ΛkRn. The results follow from the description of the volume of the projection of a polytope into ak-plane by a piecewise linear function onG(k, n). For example, the Hodge-star operator is used to obtain the volume of a polytope from its Gale transform. Also, the classification of the faces ofG(2,n) (orG(n−2,n)) imply that the largest projection within a particular combinatorial type is unique ifk=2 orn−2.
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Communicated by G. D. Chakerian
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Filliman, P. Exterior algebra and projections of polytopes. Discrete Comput Geom 5, 305–322 (1990). https://doi.org/10.1007/BF02187792
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DOI: https://doi.org/10.1007/BF02187792