A lower bound for the Quickhull convex hull algorithm that disproves the Quickhull precision conjecture

The Quickhull algorithm is a simple algorithm for constructing the convex hull of a set of n points. Quickhull is usually described for points in the plane, in which case it is defined as a divide-and-conquer algorithm, where one has a pair of points $(p,r)$ such that p and r are on the convex hull, and one then finds the point, q, farthest from the line $\overline{pr}$, which must also be on the convex hull, and then uses the triangle $(p,q,r)$ to divide the remaining points and recursively solve the resulting subproblems. It is well-known that Quickhull has a worst-case running time of $\Theta \left(n^2\right)$, but it runs much faster than this for some input distributions. In a highly cited paper, Barber, Dobkin, and Huhdanpaa conjecture that the Quickhull algorithm runs in worst-case $O(n\log h)$ time, where h is the size of the convex hull, when the input points have precision $O(\log n)$. In this paper, we give an explicit lower-bound construction that shows that, in general, the worst-case running time of the Quickhull algorithm is $\Theta \left(nh\right)$. Our lower bound proof also provides a counter-example to the Quickhull precision conjecture of Barber et al., in that we give an explicit construction of a set, S, of n points with precision $O(\log n)$ such that h is $O(\log n)$ but the worst-case running time of Quickhull on S is $\Theta \left(nh\right)$, not $O(n\log h)$.