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How hard is halfspace range searching?
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The author considers the halfspace range reporting problem: Given a finite set P of points in E/sup d/, preprocess it so that given a query halfspace gamma , the points of p intersection gamma can be reported efficiently. It is shown that, with almost linear storage, this problem can be solved substantially more efficiently than the more general simplex range searching problem. A data structure for halfspace range reporting in dimensions dor=4 is given. It uses O(n log log n) space and O (n log n) deterministic preprocessing time. The query time is also given. Results for the halfspace emptiness problem, where one only wants to know whether P intersection gamma is empty, are also presented.