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Generalized Sperner lemma and subdivisions into simplices of equal volume. (English) Zbl 0891.51013

Ivanov, A. A. (ed.) et al., Studies on topology. 8. Work collection. Sankt-Peterburg: Nauka. Zap. Nauchn. Semin. POMI. 231, 245-254 (1995).
The aim of the present note is to discuss some questions related to the following result, which apparently belongs to combinatorial geometry: A square cannot be divided into an odd number of triangles of equal area [J. Thomas, Math. Mag. 41, 187-190 (1968; Zbl 0164.51502), P. Monsky, Am. Math. Mon. 77, 161-164 (1970; Zbl 0187.19701)]. This result was transferred to the case of higher dimension by Mead: An \(n\)-dimensional cube can be divided into \(N\) simplices of equal volume iff \(N\) is divisible by \(n!\) [D. G. Mead, Proc. Am. Math. Soc. 76, 302-304 (1979; Zbl 0423.51012)]. As in the case of the square, it is not difficult to produce an example of subdivision into any number of simplices which is a multiple of \(n!,\) but to prove that this condition is necessary is much harder. In the case of arbitrary dimension the technical difficulties one has to overcome become more complicated compared to those arising in the planar case, though the general outline of the proof still works. In particular, the usual version of the Sperner Lemma concerning the triangulations is not already sufficient. Mead overcame this difficulty by using an inductive reasoning which heavily employs the fact that the coloring is defined on the whole space.
The authors use another approach and prove a more general version of the Sperner Lemma, which handles the case of an arbitrary subdivision of a polyhedron into simplices and, what can be of an independent interest, into arbitrary (convex) polyhedra. At the same time the homological meaning of the usual and the generalized version of the Sperner Lemma is established: both follow from the fact that the degree of a map of a manifold to a manifold is equal to that of the induced map of the boundary to the boundary.
For the entire collection see [Zbl 0868.00023].

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
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