Advertisement

Journal of Mathematical Sciences

, Volume 91, Issue 6, pp 3492–3498 | Cite as

Generalized Sperner lemma and subdivisions into simplices of equal volume

  • B. M. Bekker
  • N. Yu. Netsvetaev
Article

Abstract

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.

Keywords

Unit Cube Prime Integer Disjoint Interior Combinatorial Argument Transcendental Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Lang,Algebra, Addison-Wesley, Reading, Massachusetts (1965).zbMATHGoogle Scholar
  2. 2.
    P. Monsky, “On dividing a square into triangles,”Am. Math. Monthly,77, 161–164 (1970).zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    V. G. Pokrovskii, “On linear relations holding for subdivisions inton-dimensional parallelepipeds,”Mat. Zametki,37, 901–907 (1985).zbMATHMathSciNetGoogle Scholar
  4. 4.
    J. Thomas, “A dissection problem,”Math. Mag.,41, 187–190 (1968).zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    D. G. Mead, “Dissection of the hypercube into simplexes,”Proc. Am. Math. Soc.,76, 302–304 (1979).zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    J. F. Sallee, “The middle-cut triangulation of then-cube,”SIAM J. Algebraic Discrete Methods,5, 407–419 (1984).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • B. M. Bekker
  • N. Yu. Netsvetaev

There are no affiliations available

Personalised recommendations