On the connected components of the space of line transversals to a family of convex sets
Let \(A\) be a family of convex sets in \(\mathbb{R}^ d\). A \(k\)-transversal for \(A\) is a \(k\)-flat (= affine subspace of dimension \(k\)) that intersects every member of \(A\). This paper deals with the connected components of the space \(A^*\) consisting of all \(k\)-transversals of \(A\) (considered as a subspace of the affine Grassmannian of all \(k\)-flats in \(\mathbb{R}^ d\)). If \(F\) is a convex set of \(k\)-flats in \(\mathbb{R}^ d\), then \(F= A^*\) for some family of convex point sets in \(\mathbb{R}^ d\); \(F\) is then said to be represented by \(A\). A convex set \(F\) of \(k\)-flats need not be connected for \(k>0\), and a connected component of \(F\) need not be convex, as the authors show by exhibiting an example of a convex family of lines in \(\mathbb{R}^ 3\) that has a nonconvex connected component. If, however, a convex set \(F\) of \(k\)-flats in \(\mathbb{R}^ d\) is represented by some finite family of suitably separated compact convex point sets, then the authors conjecture that the connected components of \(F\) will themselves be convex. As the main result of this article, this conjecture is proved for the case \(d=3\), \(k=1\): Let \(F\) be a convex set of lines in \(\mathbb{R}^ 3\) represented by a finite family of pairwise disjoint compact convex point sets; then each connected component of \(F\) can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets.
scientific article; zbMATH DE number 764239