Weak ε-nets have basis of size O (1 /ε log (1 / ε)) in any dimension

Given a set P of n points in ℝ ^d and >0, we consider the problem of constructing weak -nets for P. We show the following: pick a random sample Q of size O(1/εlog(1/ε)) from P. Then, with constant probability, a weak ε-net of P can be constructed from only the points of Q. This shows that weak -nets in ℝ ^d can be computed from a subset of P of size O(1/εlog(1/ε)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ε. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).