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arXiv:math.MG/0211286v119Nov2002 Minkowski-type and Alexandrov-type theorems
 

Summary: arXiv:math.MG/0211286v119Nov2002
Minkowski-type and Alexandrov-type theorems
for polyhedral herissons
Victor Alexandrov
November 18, 2002
Abstract
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,
52A15.
1 Introduction
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, [1] or [5]). Moreover, H. Minkowski found natural and

  

Source: Alexandrov, Victor - Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk

 

Collections: Mathematics