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- W2079722972 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<k≤d−m) 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 k≤d−m. 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." @default.
- W2079722972 created "2016-06-24" @default.
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- W2079722972 date "2010-09-02" @default.
- W2079722972 modified "2023-10-02" @default.
- W2079722972 title "Topological transversals to a family of convex sets" @default.
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- W2079722972 doi "https://doi.org/10.1007/s00454-010-9282-z" @default.
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