Some new results on geometric transversals (Q6624172)

For any \(t\) and \(\varepsilon\in\ ]0,1[\) one may construct in \(\mathbb{R}^3\) a set \(L\) of more than \(\frac{t}{1-\varepsilon}\) lines and a finite family \(F\) of compact convex sets, each meeting an \(\varepsilon\)-fraction of \(L\), but such that the union of any \(t\) lines in \(\mathbb{R}^3\) misses at least one \(C\in F\).\N\NThere also exist 9 blue lines and 13 red lines in \(\mathbb{R}^3\) such that any convex set that meets all blue lines also meets at least one red line.\N\NA \(k\)-transversal of a family of convex sets is a \(k\)-flat meeting each member. In the space of line transversals of any family of pairwise disjoint open convex sets in \(\mathbb{R}^3\) each connected component is (topologically) acyclic. It is unknown whether this also holds for compact convex sets.\N\NThe paper also shows two more existence results for \(k\)-transversals of at least one convex set family out of several such families having special intersection or ordering structure.