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Planar polygons and hyperbolic polyhedra. (Polygones du plan et poly√®dres hyperboliques.) (French) Zbl 0758.52001
The authors consider the space of polygons in the Euclidean plane having fixed prescribed angles. These form a vector space on which the measurement of oriented area defines a quadratic form. In the case of convex polygons this construction is used to associate a convex polyhedron in a suitable hyperbolic space with the given space. The relations between the structure of this polyhedron and the geometry of the corresponding polygons are investigated. This interpretation is used for another proof of the existence of Im Hof’s hyperbolic Coxeter orthoschemes [see H.-C. Im Hof, Expo. Math. 3, 179-186 (1985; Zbl 0572.51012), and Bull. Soc. Math. Belg. Ser. A 42, No. 3, 523-545 (1990)].

MSC:
52A10 Convex sets in \(2\) dimensions (including convex curves)
52A55 Spherical and hyperbolic convexity
52B99 Polytopes and polyhedra
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