Polytope range searching and integral geometry

plexity ofsimplex range searching. We prove that the worst-case query time is 0. (n/vm), for d = 2, and more generally, 0. (nl log n)/m 1 / d ) , for d ~ 3; n is the number of points and m is the amount of stor­ age available. These bounds hold with high probability for a random point-set (from a uniform distribution) and thus are valid in the worst case as well as on the average. Interestingly, they still hold if the query remains congruent to a fixed simplex or even a fixed slab. What is the significance of these lower bounds? From a practical standpoint the news is disheartening but instructive. For the sake of il­ lustration, take d = 11: our results say that with only linear storage the query time will have to be at least 0'(nO. 9 ). To make matters worse, this quasi-linear lower bound also holds on the average, so it is un­ escapable in practice. For the query time to be lowered to, say, O(y'n), one would need g(n S ) storage, and a whopping n(n 10 ) space would be necessary if a polylogarithmic query time were desired. Things are