Reprint of: Weak ε-nets have basis of size

Given a set P of n points in R^d and ε> 0, we consider the problem of constructing weak ε-nets for P. We show the following: pick a random sample Q of size O(1/ε log(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 ε-nets in Rd can be computed from a subset of P of size O(1/ε log(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 ε-nets still have a large size (with the dimension appearing in the exponent of 1/ε).