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Geometriae Dedicata

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

Convex equipartitions: the spicy chicken theorem

  • Roman Karasev
  • Alfredo Hubard
  • Boris Aronov
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 

Notes

Acknowledgments

We thank Arseniy Akopyan, Imre Bárány, Pavle Blagojević, Sylvain Cappell, Fred Cohen, Daniel Klain, Erwin Lutwak, Yashar Memarian, Ed Miller, Gabriel Nivasch, Steven Simon, and Alexey Volovikov for discussions, useful remarks, and references. We also thank an anonymous referee for encouraging us to merge our papers and for his/her enthusiasm towards the chicken nuggets description of Corollary 1.1. Roman Karasev was supported by the Dynasty Foundation, the President’s of Russian Federation grant MD-352.2012.1, the Federal Program “Scientific and scientific-pedagogical staff of innovative Russia” 2009–2013, and the Russian government project 11.G34.31.0053. Boris Aronov and Alfredo Hubard gratefully acknowledge the support of the Centre Interfacultaire Bernoulli at EPFL, Lausanne, Switzerland. Alfredo Hubard thankfully acknowledges the support from CONACyT and from the Fondation Sciences Matheḿatiques de Paris. The research of Boris Aronov has been supported in part by a grant No. 2006/194 from the U.S.-Israel Binational Science Foundation, by NSA MSP Grant H98230-10-1-0210, and by NSF Grants CCF-08-30691, CCF-11-17336, and CCF-12-18791.

References

  1. 1.
    Agarwal, P.K., Sharir, M.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1995)zbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  4. 4.
    Aurenhammer, F., Hoffmann, F., Aronov, B.: Minkowski-type theorems and least-square clustering. Algorithmica 20, 61–72 (1998)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kaneko, A., Kano, M.: Perfect partitions of convex sets in the plane. Discret. Comput. Geom. 28(2), 211–222 (2002)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Karasev, R.N.: The genus and the category of configuration spaces. Topol. Appl. 156(14), 2406–2415 (2009)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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

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