Geometriae Dedicata

, Volume 170, Issue 1, pp 263–279 | Cite as

Convex equipartitions: the spicy chicken theorem

Original Paper

Abstract

We show that, for any prime power \(n\) and any convex body \(K\) (i.e., a compact convex set with interior) in \(\mathbb{R }^d\), there exists a partition of \(K\) into \(n\) convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov–Borsuk–Ulam theorem for convex sets in the model spaces of constant curvature.

Keywords

Equipartitions Waist Borsuk–Ulam  Ham sandwich Voronoi diagram Nandakumar–Ramana Rao conjecture Configuration space 

Mathematics Subject Classification

28A75 52A38 55M20 

References

  1. 1.
    Agarwal, P.K., Sharir, M.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  2. 2.
    Akiyama, J., Kaneko, A., Kano, M., Nakamura, G., Rivera-Campo, E., Tokunaga, S., Urrutia, J.: Radial perfect partitions of convex sets in the plane. Discrete and Computational Geometry: Japanese Conference, JCDCG’98 Tokyo, Japan, December 9–12, :Revised Papers. Akiyama, J., Kano, M., Urabe, M. (eds). Lecture Notes in Computer Science 1763, Springer 2000, 1–13 (1998)Google Scholar
  3. 3.
    Alon, N.: Splitting necklaces. Adv. Math. 63, 247–253 (1987)CrossRefMATHGoogle Scholar
  4. 4.
    Aurenhammer, F., Hoffmann, F., Aronov, B.: Minkowski-type theorems and least-square clustering. Algorithmica 20, 61–72 (1998)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bárány, I., Blagojević, P., Szűcs, A.: Equipartitioning by a convex 3-fan. Adv. Math. 223(2), 579–593 (2010)Google Scholar
  6. 6.
    Blagojević, P., Ziegler, G.: Convex equipartitions via equivariant obstruction theory. arXiv:1202.5504, (2012)Google Scholar
  7. 7.
    Borel, A., Moore, J.C.: Homology theory for locally compact spaces. Mich. Math. J. 7, 137–159 (1960)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Cohen, F.R., Taylor, L.R.: On the representation theory associated to the cohomology of configuration spaces. In: Proceedings of an International Conference on Algebraic Topology, 4–11: Oaxtepec. Contemporary Mathematics 146(1993), 91–109 (July 1991)Google Scholar
  10. 10.
    Fuks, D.B.: The mod 2 cohomologies of the braid group (In Russian). Mat. Zametki 5(2), 227–231 (1970)Google Scholar
  11. 11.
    Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Hubard, A., Aronov, B.: Convex equipartitions of volume and surface area. arXiv:1010.4611, (2010)Google Scholar
  13. 13.
    Hung, N.H.V.: The mod 2 equivariant cohomology algebras of configuration spaces. Pac. J. Math. 143(2), 251–286 (1990)CrossRefMATHGoogle Scholar
  14. 14.
    Kaneko, A., Kano, M.: Perfect partitions of convex sets in the plane. Discret. Comput. Geom. 28(2), 211–222 (2002)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Karasev, R.N.: Partitions of a polytope and mappings of a point set to facets. Discret. Comput. Geom. 34, 25–45 (2005)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Karasev, R.N.: The genus and the category of configuration spaces. Topol. Appl. 156(14), 2406–2415 (2009)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Karasev, R.N.: Equipartition of several measures. arXiv.1011.4762, (2010)Google Scholar
  18. 18.
    McCann, R.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Memarian, Y.: On Gromov’s waist of the sphere theorem. arXiv:0911.3972, (2009)Google Scholar
  20. 20.
    Nandakumar, R., Ramana Rao, N.: ‘Fair’ partitions of polygons—an introduction. arXiv:0812.2241, (2008)Google Scholar
  21. 21.
    Soberón, P.: Balanced convex partitions of measures in \(\mathbb{R}^d\). Mathematika 58(1), 71–76 (2012); first appeared as arXiv:1010.6191, (2010)Google Scholar
  22. 22.
    Steenrod, N.E.: Homology with local coefficients. Ann. Math. 44(4), 610–627 (1943)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Vasiliev, V.A.: Braid group cohomologies and algorithm complexity (In Russian). Funkts. Anal. Prilozh. 22(3), 1988, pp. 15–24. translation in. Funct. Anal. Appl. 22(3), 182–190 (1988)Google Scholar
  24. 24.
    Vasiliev, V.A.: Complements of Discriminants of Smooth Maps: Topology and Applications. Revised edition. Translations of Mathematical Monographs, 98. American Mathematical Society, (1994)Google Scholar
  25. 25.
    Villiani, C.: Optimal Transport: Old and New Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, UK (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Institute for Information Transmission Problems RASMoscowRussia
  3. 3.Laboratory of Discrete and Computational GeometryYaroslavl State UniversityYaroslavlRussia
  4. 4.Département d’InformatiqueÉcole Normale SupérieureParisFrance
  5. 5.Department of Computer Science and EngineeringPolytechnic Institute of NYUBrooklynUSA

Personalised recommendations