Topological transversals to a family of convex sets

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F$ has a \emph{topological $ρ$-transversal of index $(m,k)$} ($ρ<m$, $0<k\leq d-m$) if there are, homologically, as many transversal $m$-planes to $\mathcal F$ as $m$-planes containing a fixed $ρ$-plane in $\mathbb R^{m+k}$. 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 $k\leq d-m$. The converse is not true in general. We prove that for a family $\mathcal F$ of $ρ+k+1$ compact convex sets in $\mathbb R^d$ 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.

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