Topological transversals to a family of convex sets

Abstract

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