Discrete & Computational Geometry

, Volume 43, Issue 4, pp 876–892 | Cite as

Uneven Splitting of Ham Sandwiches

Article

Abstract

Let μ1,…,μn be continuous probability measures on ℝn and α1,…,αn∈[0,1]. When does there exist an oriented hyperplane H such that the positive half-space H+ has μi(H+)=αi for all i∈[n]? It is well known that such a hyperplane does not exist in general. The famous Ham Sandwich Theorem states that if \(\alpha_{i}=\frac{1}{2}\) for all i, then such a hyperplane always exists.

In this paper we give sufficient criteria for the existence of H for general αi∈[0,1]. Let f1,…,fn:Sn−1→ℝn denote auxiliary functions with the property that for all i, the unique hyperplane Hi with normal v that contains the point fi(v) has μi(Hi+)=αi. Our main result is that if Im f1,…,Im fn are bounded and can be separated by hyperplanes, then there exists a hyperplane H with μi(H+)=αi for all i. This gives rise to several corollaries; for instance, if the supports of μ1,…,μn are bounded and can be separated by hyperplanes, then H exists for any choice of α1,…,αn∈[0,1]. We also obtain results that can be applied if the supports of μ1,…,μn overlap.

Keywords

Partitions of masses Ham sandwich theorem Poincaré–Miranda theorem Hyperplanes Separability 

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References

  1. 1.
    Bárány, I., Matoušek, J.: Simultaneous partitions of measures by k-fans. Discrete Comput. Geom. 25, 317–334 (2001) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bespamyatnikh, S., Kirkpatrick, D., Snoeyink, J.: Generalizing ham sandwich cuts to equitable subdivisions. Discrete Comput. Geom. 24, 605–622 (2000) MATHMathSciNetGoogle Scholar
  3. 3.
    Beyer, W.A., Zardecki, A.: The early history of the Ham Sandwich Theorem. Am. Math. Mon. 111, 58–61 (2004) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Byrnes, G.B., Cairns, G., Jessup, B.: Left-overs from the Ham-Sandwich Theorem. Am. Math. Mon. 108, 246–249 (2001) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dol’nikov, V.L.: A generalization of the Ham Sandwich Theorem. Math. Notes 52, 771–779 (1992) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dumitrescu, A., Ebbers-Baumann, A., Grüne, A., Klein, R., Rote, G.: On the geometric dilation of closed curves, graphs and point sets. Comput. Geom. Theory Appl. 36, 16–38 (2006) Google Scholar
  7. 7.
    Grünbaum, B.: Partitions of mass-distributions and convex bodies by hyperplanes. Pac. J. Math. 10, 1257–1261 (1960) MATHGoogle Scholar
  8. 8.
    Guàrdia, R., Hurtado, F.: On the equipartition of plane convex bodies and convex polygons. J. Geom. 83, 32–45 (2005) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Istrăţescu, V.I.: Fixed Point Theory, Mathematics and Its Applications, vol. 7. Reidel, Dordrecht (1981) MATHGoogle Scholar
  10. 10.
    Kulpa, W.: The Poincaré–Miranda theorem. Am. Math. Mon. 104, 545–550 (1997) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Matoušek, J.: Lectures on Discrete Geometry, Graduate Texts in Mathematics, vol. 212. Springer, New York (2002) MATHGoogle Scholar
  12. 12.
    Matoušek, J.: Using the Borsuk–Ulam Theorem, corrected 2nd printing. Springer, Berlin (2008) MATHCrossRefGoogle Scholar
  13. 13.
    Mauldin, R.D. (ed.): The Scotish Book. Birkhäuser, Boston (1981) Google Scholar
  14. 14.
    Megiddo, N.: Partitioning with two lines in the plane. J. Algorithms 6, 430–433 (1985) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Miranda, C.: Un’osservazione su una teorema di Brouwer. Boll. Un. Mat. Ital. 3, 527 (1940) MathSciNetGoogle Scholar
  16. 16.
    Poincaré, H.: Sur les courbes défines par une équation différentielle IV. J. Math. Pures Appl. 85, 151–217 (1886) Google Scholar
  17. 17.
    Rado, R.: A theorem on general measure. J. Lond. Math. Soc. 21, 291–300 (1946) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Ramos, E.A.: Equipartition of mass distributions by hyperplanes. Discrete Comput. Geom. 15, 147–167 (1996) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Steinhaus, H., : A note on the Ham Sandwich Theorem. Math. Polska 11, 26–28 (1938) Google Scholar
  20. 20.
    Stojmenović, I.: Bisections and ham-sandwich cuts of convex polygons and polyhedra. In: Proceedings of the Second Canadian Conference in Computational Geometry, pp. 291–295 (1990) Google Scholar
  21. 21.
    Stojmenović, I.: Bisections and ham-sandwich cuts of convex polygons and polyhedra. Inf. Process. Lett. 38, 15–21 (1991) MATHCrossRefGoogle Scholar
  22. 22.
    Stone, A.H., Tukey, J.W.: Generalized ‘sandwich’ theorems. Duke Math. J. 9, 356–359 (1942) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Vrećica, S.T.: Tverberg’s conjecture. Discrete Comput. Geom. 29, 505–510 (2003) MATHMathSciNetGoogle Scholar
  24. 24.
    Vrećica, S.T., Živaljević, R.T.: Conical equipartitions of mass distributions. Discrete Comput. Geom. 25, 335–350 (2001) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Zindler, K.: Über konvexe Gebilde II. Monatsh. Math. 31, 25–56 (1921) MATHCrossRefGoogle Scholar
  26. 26.
    Živaljević, R.T., Vrećica, S.T.: An extension of the Ham Sandwich Theorem. Bull. Lond. Math. Soc. 22, 183–186 (1990) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany

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