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Convex equipartitions: the spicy chicken theorem

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.

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Fig. 1

Notes

  1. 1.

    Vegetarian readers are welcome to substitute the chicken filet with a peeled potato.

  2. 2.

    The reduction of Theorems 1.3–1.10 was independently discovered in [12] and [17]; it is the main contribution of these papers. The original version of Theorem 1.3 in [12] is weaker than that in [17].

  3. 3.

    The term “power” comes from Euclidean geometry. Recall that the power of a point \(p\) with respect to a circle of radius \(r\) and center \(y\), that does not contain \(p\), is \(|p-y|^2-r^2\).

References

  1. 1.

    Agarwal, P.K., Sharir, M.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1995)

    MATH  Google 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)

  3. 3.

    Alon, N.: Splitting necklaces. Adv. Math. 63, 247–253 (1987)

    Article  MATH  Google Scholar 

  4. 4.

    Aurenhammer, F., Hoffmann, F., Aronov, B.: Minkowski-type theorems and least-square clustering. Algorithmica 20, 61–72 (1998)

    Article  MATH  MathSciNet  Google 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)

  7. 7.

    Borel, A., Moore, J.C.: Homology theory for locally compact spaces. Mich. Math. J. 7, 137–159 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)

    Article  MATH  MathSciNet  Google 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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Hubard, A., Aronov, B.: Convex equipartitions of volume and surface area. arXiv:1010.4611, (2010)

  13. 13.

    Hung, N.H.V.: The mod 2 equivariant cohomology algebras of configuration spaces. Pac. J. Math. 143(2), 251–286 (1990)

    Article  MATH  Google Scholar 

  14. 14.

    Kaneko, A., Kano, M.: Perfect partitions of convex sets in the plane. Discret. Comput. Geom. 28(2), 211–222 (2002)

    Article  MATH  MathSciNet  Google 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)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Karasev, R.N.: The genus and the category of configuration spaces. Topol. Appl. 156(14), 2406–2415 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Karasev, R.N.: Equipartition of several measures. arXiv.1011.4762, (2010)

  18. 18.

    McCann, R.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Memarian, Y.: On Gromov’s waist of the sphere theorem. arXiv:0911.3972, (2009)

  20. 20.

    Nandakumar, R., Ramana Rao, N.: ‘Fair’ partitions of polygons—an introduction. arXiv:0812.2241, (2008)

  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)

    Article  MATH  MathSciNet  Google 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)

  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)

  25. 25.

    Villiani, C.: Optimal Transport: Old and New Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, UK (2009)

    Book  Google Scholar 

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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.

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Correspondence to Alfredo Hubard.

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Karasev, R., Hubard, A. & Aronov, B. Convex equipartitions: the spicy chicken theorem. Geom Dedicata 170, 263–279 (2014). https://doi.org/10.1007/s10711-013-9879-5

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Keywords

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

Mathematics Subject Classification

  • 28A75
  • 52A38
  • 55M20