Abstract
Let \(\mathcal{F}\) be a family of compact convex sets in ℝd. 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 ℝm+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 ℝd 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.
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The research of Luis Montejano is supported by CONACYT, 41340.
The research of Roman Karasev is supported by the Dynasty Foundation, the President’s of Russian Federation grant MK-113.2010.1, the Russian Foundation for Basic Research grants 10-01-00096 and 10-01-00139.
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Montejano, L., Karasev, R.N. Topological transversals to a family of convex sets. Discrete Comput Geom 46, 283–300 (2011). https://doi.org/10.1007/s00454-010-9282-z
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DOI: https://doi.org/10.1007/s00454-010-9282-z