Packing cones and their negatives in space (Q2464354)

Let \(C\) be a cone in \({\mathbb R}^3\) with base a planar convex body, and let \({\mathcal C}\) be a packing formed by translates of \(C\) and \(-C\). In 2001, W. Kuperberg proved that the density of \({\mathcal C}\) is strictly less than \(1\). The natural question arising from this fact is answered in this nice paper: the authors show that there exists an explicit constant \(c>0\) such that every packing \({\mathcal C}\) has density smaller than \(1-c\). For the sake of completeness the proof of Kuperberg's result is also sketched in the paper.