New Lower Bounds for Convex Hull Problems in Odd Dimensions

We show that in the worst case, Ω(n^d/2-1 + nlog n) sidedness queries are required to determine whether the convex hull of n points in 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 Ω(n^d/2-1) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(n^d/2) facets, the previously best lower bound for these problems is only Ω(nlog 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 ᵈ is d/2-hard in the sense of Gajentaan and Overmars.