Better Lower Bounds on Detecting Affine and Spherical Degeneracies

We show that in the worst case, Q(nd) sidedness queries are required to determine whether a set of n points in Rd is affinely degenerate, i .e . , whether it contains d + 1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing Q(nd) “collapsible” simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an Q(nd) lower bound on the number of sidedness queries required to determine the order type of a set of n points in Ed. Using similar techniques, we also show that Q(nd+l) in-sphere queries are required to decide the existence of spherical degeneracies in a set of n points in Etd.