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On the Euler characteristic of spherical polyhedra and the Euler relation

  • H. Hadwiger (a1) and P. Mani (a1)

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Let En+1, for some integer n ≥ 0, be the (n + 1)-dimensional Euclidean space, and denote by Sn the standard n–sphere in En+1, . It is convenient to introduce the (–1)-dimensional sphere , where denotes the empty set. By an i-dimensional subsphere T of Sn, i = 0 n, we understand the intersection of Sn with some (i+1)-dimensional subspace of En+1. The affine hull of T always contains, with this definition, the origin of En+1. is the unique (–1)-dimensional subsphere of Sn. By the spherical hull, sph X, of a set , we understand the intersection of all subspheres of Sn containing X. Further we set dim X: = dim sph X. The interior, the boundary and the complement of an arbitrary set , with respect to Sn, shall be denoted by int X, bd X and cpl X. Finally we define the relative interior rel int X to be the interior of with respect to the usual topology sphZ . For each (n–1)-dimensional subsphere of Sn defines two closed hemispheres of Sn, whose common boundary it is. The two hemispheres of the sphere Sº are denned to be the two one-pointed subsets of Sº. A subset is called a closed (spherical) polytope, if it is the intersection of finitely many closed hemispheres, and, if, in addition, it does not contain a subsphere of Sn. is called an i-dimensional, relatively open polytope, , or shortly an i-open polytope, if there exists a closed polytope such that dim P = i and Q = rel int P. is called a closed polyhedron, if it is a finite union of closed polytopes P1 …, Pr. The empty set is the only (–1)-dimensional closed polyhedron of Sn. We denote by the set of all closed polyhedra of Sn. is called an i-open polyhedron, for some , if there are finitely many i-open polytopes Q1 …, Qr in Sn such that , and dim . By we denote the set of all i-open polyhedra. Clearly for all , and each i-dimensional subsphere of Sn, , belongs to and to , For each i-dimensional subsphere T of Sn, set . A map is defined by , for all , and, for all .

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References

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1.Hadwiger, H., Eine Schnittrekursion fur die Eulersche Charakteristik euklidischer Polyeder mit Anwendungen innerhalb der kombinatorischen Geometrie, El. Math., 23 (1968), 121132.
2.Klee, V.The Euler characteristic in combinatorial geometry Atner. Math. Monthly, 70 (1963) 119127.
3.Lenz, H., Mengenalgebra und Eulersche Charakteristik Abh. Math. Seminar Univ. Hamburg, 34 (1970) 135147.
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On the Euler characteristic of spherical polyhedra and the Euler relation

  • H. Hadwiger (a1) and P. Mani (a1)

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