Counting Quadrics and Delaunay Triangulations and a new Convex Hull Theorem

Given a set $\cal S$ of $n$ points in three dimensions, we study the maximum numbers of quadrics spanned by subsets of points in $\cal S$ in various ways. We prove that the set of empty or enclosing ellipsoids has $\Theta(n^4)$ complexity. The same bound applies to empty general cylinders, while for empty circular cylinders a gap remains between the $\Omega(n^3)$ lower bound and the $O(n^4)$ upper bound. We also take interest in pairs of empty homothetic ellipsoids, with complexity $\Theta(n^6)$, while the specialized versions yield $\Theta(n^5)$ for pairs of general homothetic cylinders, and $\Omega(n^4)$ and $O(n^5)$ for pairs of parallel {circular} cylinders, respectively. This implies that the number of combinatorially distinct Delaunay triangulations obtained by orthogonal projections of $\cal S$ on a two-dimensional plane is $\Omega(n^4)$ and $O(n^5)$. Our lower bounds are derived from a generic geometric construction and its variants. The upper bounds result from tailored linearization schemes, in conjunction with a new result on convex polytopes which is of independent interest: In even dimensions~$d$, the convex hull of a set of $n$ points, where one half lies in a subspace of odd dimension~\mbox{$\delta > \frac{d}{2}$}, and the second half is the (multi-dimensional) projection of the first half on another subspace of dimension~$\delta$, has complexity only $\bigO{n^{\frac{d}{2}-1}}$.