Note on an inequality of Wegner (Q878066)

For \(n\geq 2\) non-overlapping unit discs in the Euclidean plane the authors study the minimum area of their convex hull. More precisely, if \(D_n\) denotes this convex hull and \(A(D_n)\) its area, the authors derive an upper bound on \(A(D_n)\) reached by a so-called Groemer packing, namely \[ A(D_n)\leq 2\sqrt{3}\cdot(n- 1)+ (2- \sqrt{4})\cdot \lceil\sqrt{12n-3}- 2\rceil+ \pi. \] They also give a number theoretic characterization of all \(n\) such that a so-called Wegner packing (which is given if and only if the lower bound on \(A(D_n)\) is attained) of \(n\) unit circles exists. They show that these numbers constitute \(95,83\dots\%\) of \(\mathbb{N}\). Finally an interesting conjecture is posed, referring to the minimum of \(A(D_n)\) in view of the perimeter of \(D_n\).