Pinning a line by balls or ovaloids in \(\mathbb R^{3}\) (Q629845)

If a line transversal \(l\) to a family \(\mathcal{F}\) of compact convex sets in \(\mathbb R^d\) cannot move without missing some \(F\in \mathcal{F}\) then is said that \(\mathcal{F}\) is a pinning of the line \(l\). The authors prove the following theorem: Let \(\mathcal{F}\) be a finite family of balls in \(\mathbb R^3\) that pin a line \(l\). If no two balls are externally tangent in a point of \(l\), then a subfamily of at most 12 of these balls pins \(l\). An ovaloid is a smooth closed surface in \(\mathbb R^3\) with strictly positive Gauss curvature everywhere. In the last part of the paper the authors show that the theorem above remains true if the balls are replaced by semialgebraic ovaloids.