Skip to main content
SpringerLink
Log in

Convex equipartitions via Equivariant Obstruction Theory

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We describe a regular cell complex model for the configuration space F(ℝd, n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. I. Arnol’d, The cohomology ring of the colored braid group, Mathematical Notes 5 (1969), 138–140.

    Article  MATH  Google Scholar 

  2. F. Aurenhammer, A criterion for the affine equivalence of cell complexes in ℝd and convex polyhedra in ℝd+1, Discrete & Computational Geometry 2 (1987), 49–64.

    Article  MATH  MathSciNet  Google Scholar 

  3. F. Aurenhammer, F. Hoffmann and B. Aronov, Minkowski-type theorems and leastsquares partitioning, in 8th Annual Symp. Comput. Geometry (SoCG), Berlin, June 1992, ACM, 1992, pp. 350–357.

  4. F. Aurenhammer, F. Hoffmann and B. Aronov, Minkowski-type theorems and leastsquares clustering, Algorithmica 20 (1998), 61–76.

    Article  MATH  MathSciNet  Google Scholar 

  5. F. Aurenhammer and R. Klein, Voronoi diagrams, in Handbook of Computational Geometry, Noth-Holland, 2000, pp. 201–290.

  6. D. Ayala and R. Hepworth, Configuration spaces and Θn, Proceedings of the American Mathematical Society, to appear. http://arxiv.org/abs/1202.2806.

  7. I. Bárány, P. V. M. Blagojević and A. Szűcs, Equipartitioning by a convex 3-fan, Advances in Mathematics 223 (2010), 579–593.

    Article  MATH  MathSciNet  Google Scholar 

  8. I. Basabe, J. González, Yu. B. Rudyak and D. Tamaki, Higher topological complexity and homotopy dimension of configuration spaces of spheres, preprint, http://arxiv.org/abs/1009.1851.

  9. A. Björner, Subspace arrangements, in Proceedings of the First European Congress of Mathematics (Paris 1992), Vol. I, Birkhäuser, Basel, 1994, pp. 321–370.

    Google Scholar 

  10. A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Second edn., Encyclopedia of Mathematics and its Applications, Vol. 46, Cambridge University Press, Cambridge, 1993.

    MATH  Google Scholar 

  11. A. Björner and G. M. Ziegler, Combinatorial stratification of complex arrangements, Journal of the American Mathematical Society 5 (1992), 105–149.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. V. M. Blagojević and A. S. Dimitrijević Blagojević, Using equivariant obstruction theory in combinatorial geometry, Topology and its Applications 154 (2007), 2635–2655.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. V. M. Blagojević, W. Lück and G. M. Ziegler, Equivariant topology of configuration spaces, preprint, http://arxiv.org/abs/1207.2852.

  14. A. Borel and J. Moore, Homology theory for locally compact spaces, The Michigan Mathematical Journal 7 (1960), 137–159.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. E. Chisholm, k-regular mappings of 2n-dimensional euclidean space, Proceedings of the American Mathematical Society 74 (1979), 187–190.

    MATH  MathSciNet  Google Scholar 

  16. F. R. Cohen and J. E. Connett, A coincidence theorem related to the Borsuk-Ulam theorem, Proceedings of the American Mathematical Society 44 (1974), 218–220.

    Article  MATH  MathSciNet  Google Scholar 

  17. F. R. Cohen and D. Handel, k-regular embeddings of the plane, Proceedings of the American Mathematical Society 72 (1978), 201–204.

    MATH  MathSciNet  Google Scholar 

  18. G. E. Cooke and R. L. Finney, Homology of Cell Complexes, Mathematical Notes, Princeton University Press, Princeton, NJ, 1967.

    MATH  Google Scholar 

  19. C. de Concini and M. Salvetti, Cohomology of Coxeter groups and Artin groups, Mathematical Research Letters 7 (2000), 213–232.

    MATH  MathSciNet  Google Scholar 

  20. P. Deligne, Les immeubles des groupes de tresses généralisés, Inventiones Mathematicae 17 (1972), 273–302.

    MATH  MathSciNet  Google Scholar 

  21. T. tom Dieck, Transformation Groups, de Gruyter Studies in Mathematics, Vol. 8, Walter de Gruyter, Berlin, 1987.

    MATH  Google Scholar 

  22. A. Dold, Simple proofs of some Borsuk-Ulam results, Contemporary Mathematics 19 (1983), 65–69.

    MATH  MathSciNet  Google Scholar 

  23. L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current Developments in Mathematics, 1997, Cambridge, MA, Internatinal Press, Boston, MA, pp. 65–126. Updated and corrected version, Sept. 2001, http://math.berkeley.edu/~evans/Monge-Kantorovich.survey.pdf.

    Google Scholar 

  24. R. Fox and L. Neuwirth, The braid groups, Mathematica Scandinavica 10 (1962), 119–126.

    MATH  MathSciNet  Google Scholar 

  25. D. B. Fuks, Cohomologies of the braid groups mod 2, Functional Analysis and its Applications 4 (1970), 143–151.

    MATH  Google Scholar 

  26. D. Geiss, R. Klein, R. Penninger and G. Rote, Optimally solving a transportation problem using Voronoi diagrams, Computational Geometry, Theory and Applications, special issue for the 28-th European Workshop on Computational Geometry (EuroCG’12), Vol. 46, 2013, pp. 1009–1016. http://arxiv.org/abs/1206.3057

    MATH  MathSciNet  Google Scholar 

  27. I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston, 1994.

    MATH  Google Scholar 

  28. C. Giusti and D. Sinha, Fox-Neuwirth cell structures and the cohomology of the symmetric group, pub. del Centro de Giorgi, to appear. http://arxiv.org/abs/1110.4137.

  29. M. Goresky and R. D. MacPherson, Stratified Morse Theory, Ergebnisse der Mathematikund ihrer Grenzgebiete, Vol. 14, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  30. A. Hubard and B. Aronov, Convex equipartitions of volume and surface area, preprint, http://arxiv.org/abs/1306.2741. Published as R. Karaser, A.Hubard and B. Arnov, Convex Equipartitions: The Spicy Chicken Theorem, Geometriae Dedicata, online July 2013.

  31. H. Joris, C. Oestreicher and J. Steinig, The greatest common divisor of certain sets of binomial coefficients, Journal of Number Theory 21 (1985), 101–119.

    MATH  MathSciNet  Google Scholar 

  32. L. V. Kantorovich, A new method of solving some classes of extremal problems, Doklady Akademii Nauk SSSR 28 (1940), 211–214.

    Google Scholar 

  33. G. Kaplan and D. Levy, GCD of truncated rows in Pascal’s triangle, Integers. Electronic Journal of Combinatorial Number Theory 4 (2004), p. #A14. http://www.emis.de/journals/INTEGERS/papers/e14/e14.pdf.

    MathSciNet  Google Scholar 

  34. R. N. Karasev, Equipartition of several measures, preprint, http://arxiv.org/abs/1011.4762.

  35. M. Lazard, Sur les groupes de Lie formels à un paramètre, Bulletin de la Société Mathématique de France 83 (1955), 251–274.

    MATH  MathSciNet  Google Scholar 

  36. J. Matoušek, Using the Borsuk-Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry, Universitext, Springer, Heidelberg 2003; second printing, 2008.

    MATH  Google Scholar 

  37. J. Milnor, Collected Papers, Vol. III: Differential Topology, American Mathematical Society, Providence, RI, 2007.

    Google Scholar 

  38. J. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, Vol. 76, Princeton University Press, Princeton, NJ, 1974.

    MATH  Google Scholar 

  39. H. Minkowski, Allgemeine Lehrsätze über konvexe Polyeder, Nachrichten von der Könglichen Gesellschaft der Wissenschaften zu Göttingen (1897), 198–219. Reprinted in Gesammelte Abhandlungen II (Leipzig and Berlin, 1911) 103–121.

  40. H. Minkowski, Volumen und Oberfläche, Mathematische Annalen 57 (1903), 447495. Reprinted in Gesammelte Abhandlungen II (Leipzig and Berlin, 1911) 230276.

    MathSciNet  Google Scholar 

  41. J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, Menlo Park, CA, 1984.

    MATH  Google Scholar 

  42. R. Nandakumar, “Fair” partitions. Blog entry, http://nandacumar.blogspot.de/2006/09/cutting-shapes.html, September 28, 2006.

  43. R. Nandakumar and N. Ramana Rao, Fair partitions of polygons: An elementary introduction, Indian Academy of Sciences. Proceedings. Mathematical Sciences 122 (2012), 459–467.

    MATH  MathSciNet  Google Scholar 

  44. S. P. Novikov, Homotopy properties of Thom complexes, Matematicheskiĭ Sbornik 57 (1962), 407–442.

    Google Scholar 

  45. V. V. Prasolov, Elements of Homology Theory, Graduate Studies in Mathematics, Vol. 81, American Mathrmatical Society, Providence, RI, 2007.

    MATH  Google Scholar 

  46. B. Ram, Common factors of \(\tfrac{{n!}} {{m!(n - m)!}}(m = 1,2,...n - 1)\), Journal of the Indian Mathematical Club 1 (1909), 39–43.

    MATH  Google Scholar 

  47. M. Salvetti, Topology of the complement of real hyperplanes in ℂN, Inventiones Mathematicae 88 (1987), 603–618.

    MATH  MathSciNet  Google Scholar 

  48. D. Siersma and M. van Manen, Power diagrams and applications, preprint, http://arxiv.org/abs/math/0508037.

  49. C. Soulé, Secant varieties and successive minima, Journal of Algebraic Geometry 13 (2004), 323–341.

    MATH  MathSciNet  Google Scholar 

  50. V. A. Vassiliev, Braid group cohomologies and algorithm complexity, Functional Analysis and its Applications 22 (1988), 182–190.

    MathSciNet  Google Scholar 

  51. V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and Applications, Translations of Mathematical Monographs, Vol. 98, American Mathematical Society, Providence, RI, 1992.

    Google Scholar 

  52. C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  53. G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Springer-Verlag, New York, 1995; seventh updated printing 2007.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematički Institut SANU, Knez Mihailova 36, 11001, Beograd, Serbia

    Pavle V. M. Blagojević

  2. Institut für Mathematik, FU Berlin, Arnimallee 2, 14195, Berlin, Germany

    Pavle V. M. Blagojević & Günter M. Ziegler

Authors
  1. Pavle V. M. Blagojević
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Günter M. Ziegler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavle V. M. Blagojević.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels. The first author was also supported by the grant ON 174008 of the Serbian Ministry of Education and Science.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Blagojević, P.V.M., Ziegler, G.M. Convex equipartitions via Equivariant Obstruction Theory. Isr. J. Math. 200, 49–77 (2014). https://doi.org/10.1007/s11856-014-1006-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-014-1006-6

Keywords

Search

Navigation