Stabbing Delaunay tetrahedralizations (Q705126)

In this remarkable paper a Delaunay tetrahedralization of \(n\) vertices is exhibited for which a straight line passes through the interior of \(\Theta(n^2)\) tetrahedra. This solves an open problem of Amenta [see \textit{N. Amenta} and \textit{G. Ziegler}, Contemp. Math. 223, 57--90 (1999; Zbl 0916.90205)], who asked whether a line can stab more than \(O(n)\) tetrahedra. In higher dimensions \(d\) the answer is \(\Theta(n^{\lceil d/2\rceil})\). As a consequence, it follows that a two-dimensional slice of a \(d\)-dimensional \(n\)-vertex polytope can have \(\Theta(n^{\lceil d/2\rceil})\) facets. This last result of Amenta and Ziegler (see paper above) has been simplified by the author.