Topological transversals to a family of convex sets (Q635748)

For a family \(\mathcal{F}\) of compact convex sets in \(\mathbb{R}^d\) and an integer number \(m\), an \(0<m<d\), \(m\)-transversal, \(\mathcal{T}_m(\mathcal{F})\), is defined as the set of all \(m\)-planes intersecting every \(A\in\mathcal{F}\). It is defined that a family \(\mathcal{F}\) has a topological \(\varrho\)-transversal of index \((m,k)\), \(\varrho<m\), \(0<k<d-m\), if there are homologically as many transversal \(m\)-planes as \(m\)-planes containing a fixed \(\varrho\)-plane in \(\mathbb{R}^{m+k}\). The authors obtain that if a family \(\mathcal{F}\) of \(\varrho+k+1\) compact convex sets in \(\mathbb{R}^{d}\) has a topological \(\varrho\)-transversal of index \((m,k)\), then the family \(\mathcal{F}\) also has an ordinary \(\varrho\)-transversal. This result and its generalization imply transversal analogues of the colorful Helly theorem and some other geometric applications.