Reporting points in halfspaces

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 d>or=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.<>