Abstract: 
A finite set N ⊃ Rd is a weak εnet for an npoint set X ⊃ Rd (with respect to convex sets) if N intersects every convex set K with K ∩ X ≥ εn. We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al. [7], that every point set X in Rd admits a weak εnet of cardinality O(εd polylog(1/ε)). Moreover, for a number of special point sets (e.g., for points on the moment curve), our method gives substantially better bounds. The construction yields an algorithm to construct such weak εnets in time O(n ln(1/ε)). We also prove, by a different method, a nearlinear upper bound for points uniformly distributed on the (d  1)dimensional sphere.
