Simplex range reporting on a pointer machine

We give a lower bound on the following problem, known as simplex range reporting: Given a collection P of n points in d-space and an arbitrary simplex q, find all the points in P boolean AND q. It is understood that P is fixed and can be preprocessed ahead of time, while q is a query that must be answered on-line. We consider data structures for this problem that can be modeled on a pointer machine and whose query time is bounded by O(n(delta) + r), where r is the number of points to be reported and delta is an arbitrary fixed real. We prove that any such data structure of that form must occupy storage Omega(n(d(1-delta)-epsilon)), for any fixed epsilon > 0. This lower bound is tight within a factor of n(epsilon).