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We present a new technique for half-space and simplex range query using &Ogr;(n) space and &Ogr;(na) query time, where a d(d-1)/d(d-1) + 1 + &ggr; for all dimensions d ≥ 2 and &ggr; 0. These bounds are better than those previously published for all d ≥ 2. The technique uses random sampling to build a partition-tree structure. We introduce the concept of an &egr;-net for an abstract set of ranges to describe the desired result of this random sampling and give necessary and sufficient conditions that a random sample is an &egr;-net with high probability. We illustrate the application of these ideas to other range query problems.