Product range spaces, sensitive sampling, and derandomization

We introduce the concept of a sensitive ε-approximation, and use it to derive a more efficient algorithm for computing ε-nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VC-dimensional case. This generalizes and simplifies results from previous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(n log³ n) deterministic algorithm for computing the diameter of an n-point set in 3-dimensional space.