We derive two results from Clarkson's randomized algorithmfor linear programming in a fixed dimension d.The first is a simple general method that reduces theproblem of answering linear programming queries tothe problem of answering halfspace range queries. Forexample, this yields a randomized data structure withO(n) space and O(n1\Gamma1=bd=2c2O(logn)) query time forlinear programming on n halfspaces (d ? 3). The secondresult is a simpler proof of the following: a...