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On Schläfli’s reduction formula. (English) Zbl 0717.52011

The theory of orthoschemes is helpful since orthoschemes are very suitable basic objects in polyhedral geometry and in particular for the problem of calculating volume of non-Euclidean polytopes. An orthoscheme is a spherical simplex also called orthogonal simplex. It is a generalization of a right triangle to higher dimensions. The well known volume formula for a spherical triangle was generalized by L. Schläfli [Theorie der vielfachen Kontinuität; in: Gesammelte mathematische Abhandlungen I, Birkhäuser Basel (1949; Zbl 0035.219)] to spherical simplices of even dimensions. That formula reduces the volume of these ones to the volume of simplices of lower dimensions and is called Schläfli’s reduction formula. It was extended by M. H. Poincaré [Sur la généralisation d’un théorème élémentaire de géométrie. C. R. Acad. Sci. Paris 140, 113-117 (1905)] to the hyperbolic case and summarized by E. Peschl for elliptic (spherical) and hyperbolic spaces. Of course this formula is valid for orthoschemes.
The author considers a more general class of hyperbolic polytopes, the so called (complete) orthoschemes of degree d, \(0\leq d\leq 2\) with \(d=0\) for a usual orthoscheme. An orthoscheme of degree \(d=1\) or \(d=2\) can easily be explained in the projective model for the d-dimensional hyperbolic space: Allowing one or both of the principal vertices of the orthoscheme to lie outside the fundamental quadric and cutting off those ideal verticel by means of their polar hyperplanes the remaining d times truncated orthoscheme is called a complete one of degree d. These polytopes arise e.g. as a particular class of fundamental polytopes in the classification problem for hyperbolic Coxeter groups [cf. H. S. M. Coxeter, Ann. Math., II. Ser. 35, 588-621 (1934; Zbl 0010.01101)]; H.-C. Im Hof, Expo. Math. 3, 179-186 (1985; Zbl 0572.51012); Eh. B. Vinberg, Hyperbolic Reflection Groups. Russ. Math. Surv. 40, No. 1, 31-75 (1985), translation from Usp. Mat. Nauk 40, No. 1(241), 29-66 (1985; Zbl 0579.51015)]. The author establishes a generalized reduction formula which holds for even dimensional complete orthoschemes giving Schläfli’s formula for \(d=0\). Using Coxeter graphs the generalized reduction formula can be proved by induction. At the end the graphs and volumes of all complete Coxeter orthoschemes of degree \(d\in \{0,1,2\}\) and even dimension \(\geq 4\) are given. As H.-C. Im Hof has shown that there exist only such orthoschemes up to dimension 9 and in each such dimension \(\geq 4\) there are finitely many examples the list is complete for all dimensions \(\geq 4\).
Reviewer: J.Böhm

MSC:

52B12 Special polytopes (linear programming, centrally symmetric, etc.)
51M20 Polyhedra and polytopes; regular figures, division of spaces
51M10 Hyperbolic and elliptic geometries (general) and generalizations
52A38 Length, area, volume and convex sets (aspects of convex geometry)
51F15 Reflection groups, reflection geometries
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References:

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