Discrete & Computational Geometry

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

Topological transversals to a family of convex sets

  • L. MontejanoEmail author
  • R. N. Karasev


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.


Common transversal The Helly theorem The Schubert calculus 


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© 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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