New constructions of weak epsilon-nets

A finite set ? ? R^d is a <i>weak e-net</i> for an <i>n</i> -point set <i>X</i> ? <b>R</b> ^<i>d</i> (with respect to convex sets) if <i>N</i> intersects every convex set <i>K</i> with | <i>K</i> n <i>X</i> |= en. We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al. [7], that every point set <i>X</i> in <b>R</b> ^<i>d</i> admits a weak e-net of cardinality <i>O</i> (e ^<i>-d</i> polylog(1/e)). 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 eps-nets in time <i>O</i> ( <i>n</i> ln(1e)). We also prove, by a different method, a near-linear upper bound for points uniformly distributed on the (d--1)-dimensional sphere.