Topological transversals to a family of convex sets

Let F be a family of compact convex sets in R. We say that 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 F as m-planes containing a fixed ρ-plane in R. Clearly, if F has a ρ-transversal plane, then 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 F of ρ+k+1 compact convex sets in R 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.