Discrete & Computational Geometry

, Volume 43, Issue 2, pp 477–481 | Cite as

Equipartition of a Measure by (Z p ) k -Invariant Fans

  • R. N. KarasevEmail author


We prove a result about partitioning an absolutely continuous measure in ℝ d into 2d equal parts by a system of cones with common vertex, where d is an odd prime power. The proof is topological and based on the calculation of the equivariant Euler class of a certain vector bundle.


Knaster’s problem Equivariant topology Inscribing Measure partition 


  1. 1.
    Bárány, I., Matoušek, J.: Simultaneous partitions of measures by k-fans. Discrete Comput. Geom. 25(3), 317–334 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Grünbaum, B.: Partitions of mass-distributions and of convex bodies by hyperplanes. Pac. J. Math. 10, 1257–1261 (1960) zbMATHGoogle Scholar
  3. 3.
    Hsiang, W.Y.: Cohomology Theory of Topological Transformation Groups. Springer, Berlin (1975) zbMATHGoogle Scholar
  4. 4.
    Karasev, R.N.: Dual theorems on central points and their generalizations. Sb. Math. 199(10), 1459–1479 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Karasev, R.N.: Inscribing a regular crosspolytope. Math. Notes (in Russian) (to appear) Google Scholar
  6. 6.
    Knaster, B.: Problem 4. Colloq. Math. 30, 30–31 (1947) Google Scholar
  7. 7.
    Luke, G., Mishchenko, A.S.: Vector Bundles and their Applications. Springer, Berlin (1998) zbMATHGoogle Scholar
  8. 8.
    Makeev, V.V.: Partitioning space in six parts. Vestn. Leningr. State Univ. 2, 31–34 (1988) (in Russian) MathSciNetGoogle Scholar
  9. 9.
    Makeev, V.V.: Inscribed and circumscribed polyhedra of a convex body. Math. Notes 55(4), 423–425 (1994) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Milnor, J., Stasheff, J.: Characteristic Classes. Princeton University Press, Princeton (1974) zbMATHGoogle Scholar
  11. 11.
    Steinhaus, H.: Sur la division des ensembles de l’espaces par les plans et des ensembles plans par les cercles. Fund. Math. 33, 245–263 (1945) zbMATHMathSciNetGoogle Scholar
  12. 12.
    Stone, A.H., Tukey, J.W.: Generalized ‘sandwich’ theorems. Duke Math. J. 9, 356–359 (1942) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Volovikov, A.Y.: A theorem of Bourgin–Yang type for ℤpn-action. Russ. Acad. Sci. Math. Sb. 76(2), 361–387 (1993) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Volovikov, A.Y.: Coincidence points of maps of ℤpn-spaces. Izv. Math. 69(5), 913–962 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Vrećica, S.T., Živaljević, R.T.: Conical equipartitions of mass distributions. Discrete Comput. Geom. 25, 335–350 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Živaljević, R.: Topological Methods. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC, Boca Raton (2004) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Dept. of MathematicsDolgoprudnyRussia

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