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. Bibliography: 6 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 245–254.