We show that in the worst case, (n d) sidedness queries are required to determine whether a set of n points in IR d is a nely degenerate, i.e., whether it contains d +1points 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 (n d) \collapsible " simplices, any oneof which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, wehave an (n d)lower bound on the number of sidedness queries required to determine the order type of a set of n points in IR d. Using similar techniques, we alsoshowthat (n d+1) in-sphere queries are required to decide the existence of spherical degeneracies in a set of n points in IR d. 1
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