On the dissection of simplices into orthoschemes. (English) Zbl 0780.52016

In this paper decompositions of polytopes in an \(n\)-dimensional Euclidean space into orthoschemes are considered. Because every polytope can be decomposed into simplices it suffices to investigate the set of simplices. H. Hadwiger [“Vorlesungen über Inhalt, Oberfläche und Isoperimetrie” (1957; Zbl 0078.357)] conjectured that every Euclidean simplex can be decomposed into orthoschemes. For dimensions \(n\leq 4\) several authors had confirmed this conjecture (cf. H. Chr. Lenhard, the reviewer, H. Schulow, A. B. Kharazishvili). But it seems difficult to generalize their methods for higher dimensions.
A second question is to give the decomposition number for the sets of Euclidean \(n\)-dimensional simplices. This number \(N(n)\) is the least natural number such that each \(n\)-dimensional simplex can be decomposed into \(N(n)\) or less orthoschemes. We have \(N(2)=2\) and \(N(3)=12\) (cf. H. Chr. Lenhard, the reviewer). For \(n=4\) A. B. Kharazichvili [Soobshch. Akad. Nauk Gruz. SSR 88, 33-36 (1978; Zbl 0382.52004)] has shown \(N(4)\leq 730\); the reviewer and H. Schwulow [Wiss. Z. Friedrich-Schiller-Univ. Jena. Math.-Naturwiss. Reihe 31, 545-555 (1982; Zbl 0501.51014)] had received \(N(4)\leq 640\).
Here the author gives some general propositions on dissecting an \(n\)- dimensional simplex into two subsimplices. Using a helpful graph- theoretical idea of M. Fiedler [Časopis Pešt. Mat. 79, 297- 320 (1954; Zbl 0059.140)] it can be shown that \(N(4)\leq 500\). Many principal cases must be discussed. This method can be generalized for higher dimensions. For \(n=5\) the author shows this in another paper [Diss. Univ. Jena 1993; see also the author’s paper which is to appear in Beitr. Algebra Geom. 35, No. 1, 1-11 (1994)].
Reviewer: J.Böhm (Jena)


52B45 Dissections and valuations (Hilbert’s third problem, etc.)
51M20 Polyhedra and polytopes; regular figures, division of spaces
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