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.
Similar content being viewed by others
Notes
Vegetarian readers are welcome to substitute the chicken filet with a peeled potato.
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
Agarwal, P.K., Sharir, M.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1995)
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)
Alon, N.: Splitting necklaces. Adv. Math. 63, 247–253 (1987)
Aurenhammer, F., Hoffmann, F., Aronov, B.: Minkowski-type theorems and least-square clustering. Algorithmica 20, 61–72 (1998)
Bárány, I., Blagojević, P., Szűcs, A.: Equipartitioning by a convex 3-fan. Adv. Math. 223(2), 579–593 (2010)
Blagojević, P., Ziegler, G.: Convex equipartitions via equivariant obstruction theory. arXiv:1202.5504, (2012)
Borel, A., Moore, J.C.: Homology theory for locally compact spaces. Mich. Math. J. 7, 137–159 (1960)
Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)
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)
Fuks, D.B.: The mod 2 cohomologies of the braid group (In Russian). Mat. Zametki 5(2), 227–231 (1970)
Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)
Hubard, A., Aronov, B.: Convex equipartitions of volume and surface area. arXiv:1010.4611, (2010)
Hung, N.H.V.: The mod 2 equivariant cohomology algebras of configuration spaces. Pac. J. Math. 143(2), 251–286 (1990)
Kaneko, A., Kano, M.: Perfect partitions of convex sets in the plane. Discret. Comput. Geom. 28(2), 211–222 (2002)
Karasev, R.N.: Partitions of a polytope and mappings of a point set to facets. Discret. Comput. Geom. 34, 25–45 (2005)
Karasev, R.N.: The genus and the category of configuration spaces. Topol. Appl. 156(14), 2406–2415 (2009)
Karasev, R.N.: Equipartition of several measures. arXiv.1011.4762, (2010)
McCann, R.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995)
Memarian, Y.: On Gromov’s waist of the sphere theorem. arXiv:0911.3972, (2009)
Nandakumar, R., Ramana Rao, N.: ‘Fair’ partitions of polygons—an introduction. arXiv:0812.2241, (2008)
Soberón, P.: Balanced convex partitions of measures in \(\mathbb{R}^d\). Mathematika 58(1), 71–76 (2012); first appeared as arXiv:1010.6191, (2010)
Steenrod, N.E.: Homology with local coefficients. Ann. Math. 44(4), 610–627 (1943)
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)
Vasiliev, V.A.: Complements of Discriminants of Smooth Maps: Topology and Applications. Revised edition. Translations of Mathematical Monographs, 98. American Mathematical Society, (1994)
Villiani, C.: Optimal Transport: Old and New Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, UK (2009)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-013-9879-5
Keywords
- Equipartitions
- Waist
- Borsuk–Ulam
- Ham sandwich
- Voronoi diagram
- Nandakumar–Ramana Rao conjecture
- Configuration space