We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space H^n may only exist if n≤ 14. We also provide a family of such polyhedra of dimensions n=3,4,...,8 . We prove that for n=3,4 the members of this family have the minimal total number of hyperfaces and cusps among all hyperbolic right-angled polyhedra of the corresponding dimension. This fact is used in the proof of the main result.
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Potyagailo et al. (2005) studied this question.
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