Minkowski-type and Alexandrov-type theorems
for polyhedral herissons
November 18, 2002
Classical H. Minkowski theorems on existence and uniqueness of con-
vex polyhedra with prescribed directions and areas of faces as well as the
well-known generalization of H. 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.
Key words: convex polyhedron, polyhedral surface, polyhedral hedge-
hog, equipment, virtual polytope, polygon, Cauchy lemma, open mapping.
2000 Mathematics Subject Classification: 52B10, 52C25, 52B70, 52A38,
H. Minkowski proved that a convex polyhedron is uniquely (up to translation)
determined by the areas of its faces and the unit outward normal vectors to the
faces (see, for example,  or ). Moreover, H. Minkowski found natural and