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This paper establishes a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. 2000 Mathematics subject classification: primary 18C05, 52A01; secondary 08A35, 08A05, 18A35.doi:10.1017/s1446788708000554 fatcat:e2zbzmxbbvaqlkbzyxycko2c54