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Discrete & Computational Geometry

, Volume 46, Issue 2, pp 283–300 | Cite as

Topological transversals to a family of convex sets

  • L. Montejano
  • R. N. Karasev
Article

Abstract

Let \(\mathcal{F}\) be a family of compact convex sets in ℝ d . We say that \(\mathcal{F}\) has a topological ρ-transversal of index (m,k) (ρ<m, 0<kdm) if there are, homologically, as many transversal m-planes to \(\mathcal{F}\) as m-planes containing a fixed ρ-plane in ℝ m+k .

Clearly, if \(\mathcal{F}\) has a ρ-transversal plane, then \(\mathcal{F}\) has a topological ρ-transversal of index (m,k), for ρ<m and kdm. The converse is not true in general.

We prove that for a family \(\mathcal{F}\) of ρ+k+1 compact convex sets in ℝ d a topological ρ-transversal of index (m,k) implies an ordinary ρ-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik–Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by Bárány and Lovász, to obtain some geometric consequences.

Keywords

Common transversal The Helly theorem The Schubert calculus 

References

  1. 1.
    Arocha, J., Bracho, J., Montejano, L., Oliveros, D., Strausz, R.: Separoids; their categories and a Hadwiger-type theorem for transversals. Discrete Comput. Geom. 27(3), 377–385 (2002) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Arocha, J., Bárány, I., Bracho, J., Fabila, R., Montejano, L.: Very colorful theorems. Discrete Comput. Geom. 42(2), 142–184 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40, 141–152 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bracho, J., Montejano, L.: Helly type theorems on the homology of the space of transversals. Discrete Comput. Geom. 27(3), 387–393 (2002) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bracho, J., Montejano, L., Oliveros, D.: The topology of the space of transversals through the space of configurations. Topol. Appl. 120(12), 92–103 (2002) MathSciNetGoogle Scholar
  6. 6.
    Cappell, S.E., Goodman, J.E., Pach, J., Pollack, R., Sharir, M., Wenger, R.: Common tangents and common transversals. Adv. Math. 106, 198–215 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chern, S.S.: On the multiplication in the characteristic ring of a sphere bundle. Ann. Math. 49, 362–372 (1948) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Dol’nikov, V.L.: Transversals of families of sets in ℝn and a connection between the Helly and Borsuk theorems. Sb. Math. 79(1), 93–107 (1994) MathSciNetGoogle Scholar
  9. 9.
    Hiller, H.L.: On the cohomology of real grassmannians. Trans. Am. Math. Soc. 257(2), 521–533 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hiller, H.L.: On the height of the first Stiefel–Whitney class. Proc. Am. Math. Soc. 79(3), 495–498 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Karasev, R.N.: Theorems of Borsuk-Ulam type for flats and common transversals of families of convex compact sets. Sb. Math. 200(10), 1453–1471 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Annals of Mathematical Studies, vol. 76. Princeton University Press, Princeton (1974) zbMATHGoogle Scholar
  13. 13.
    Montejano, L.: Transversals, topology and colorful geometric results. In: Intuitive Geometry. Bolyai Society Mathematical Studies. Springer, Berlin (2010, to appear). Volume dedicated to László Fejes Tóth’s memory Google Scholar
  14. 14.
    Pontryagin, L.S.: Characteristic cycles on differential manifolds. Trans. Am. Math. Soc. 32, 149–218 (1950) Google Scholar
  15. 15.
    Porteous, I.R.: Simple singularities of maps. In: Wall, C.T.C. (ed.) Proc. Liverpool Singularity Symposium I. Springer Lecture Notes in Mathematics, vol. 192, pp. 286–307. Springer, Berlin (1971) CrossRefGoogle Scholar
  16. 16.
    Živaljević, R.: Topological Methods. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC, Boca Raton (2004) Google Scholar
  17. 17.
    Živaljević, R.T., Vrećica, S.T.: An extension of the ham sandwich theorem. Bull. Lond. Math. Soc. 22, 183–186 (1990) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacionál Autónoma de MéxicoMexico D.F.Mexico
  2. 2.Dept. of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia

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