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Minkowski-type and Alexandrov-type theorems for polyhedral herissons. (English) Zbl 1110.52013
Summary: The classical Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of the Minkowski uniqueness theorem due to A. D. Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be described as polyhedra with injective spherical image.
MSC:
52B10 Three-dimensional polytopes
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
52B70 Polyhedral manifolds
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
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