New Lower Bounds for Convex Hull Problems in Odd Dimensions

We show that in the worst case, $\Omega(n^{\ceil{d/2}-1} + n\log n)$ sidedness queries are required to determine whether the convex hull of n points in $\Real^d$ is simplicial or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with $\Omega(n^{\ceil{d/2}-1})$ degenerate facets. While it has been known for several years that d-dimensional convex hulls can have $\Omega(n^{\floor{d/2}})$ facets, the previously best lower bound for these problems is only $\Omega(n\log n)$. Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in $\Real^d$ is $\ceil{d/2}$\SUM-hard in the sense of Gajentaan and Overmars.