In an earlier paper, Romanowska, Ślusarski and Smith described 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. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.
