Cutting a set of disks by a line with leaving many intact disks in both sides (Q1976330)

Let \(\mathcal F\) be a family of \(n\) disjoint disks and \(L\) be a line in the plane. Let \(a(L)\) and \(b(L)\) be the number of disks in \(\mathcal F\) on the two sides of \(L\) that are disjoint from \(L\). The authors prove the following three statements. For any family \(\mathcal F\) of \(n\) disjoint unit disks, there is a line \(L\) such that \(\min (a(L),\;b(L))=n/2-o(n)\). There is a family of infinitely many disjoint disks in the plane such that for every line \(L\), \(\min (a(L),\;b(L))\leq 1\). For any family \(\mathcal F\) of \(n\) disjoint unit disks in the plane there exists a circle \(C\) such that its interior contains more than \({1\over 4} n-o(n)\) disks and its exterior contains more than \({1\over 4} n-o(n)\) disks. The authors also formulate the following question. What is the maximum value \(c\) that replaces \({1\over 4}\) in the last statement?