Generalized Sperner lemma and subdivisions into simplices of equal volume

A generalization of the well-known Sperner lemma is suggested, which covers the case of arbitrary subdivisions of (convex) polyhedra into (convex) polyhedra. It is used for giving a new proof of the Thomas–Monsky–Mead theorem saying that the $n$-cube can be subdivided into $N$ simplices of equal volume if and only if $N$ is divisible by $n!$. Some new related results are announced. Bibl. 6 titles.