Stabbing Delaunay tetrahedralizations

A Delaunay tetrahedralization of n vertices is exhibited for which a straight line can pass through the interiors of Θ(n 2) tetrahedra. This solves an open problem of Nina Amenta, who asked whether a line can stab more than O(n) tetrahedra. The construction generalizes to higher dimensions: in d dimensions, a line can stab the interiors of Θ(n ⌈d/2 ⌉ ) Delaunay d-simplices. The relationship between a Delaunay triangulation and a convex polytope yields another result: a two-dimensional slice of a d-dimensional n-vertex polytope can have Θ(n ⌊d/2 ⌋ ) facets. This last result was first demonstrated by Amenta and Ziegler, but the construction given here is simpler and more intuitive. Supported in part by the National Science Foundation under Awards ACI-9875170, CMS-9980063, CCR-0204377, and EIA-9802069, and in part by a gift from the Okawa Foundation. The views and conclusions contained in this document are those of the author. They are not endorsed by, and do not necessarily reflect the position or policies of, the U. S. Government or other sponsors