Before revision:  We offer two alternative schemes that yield significantly smaller sets, and two related results, as follows: (A)Let be a sample of size ] \tldO(d ^2 ^{-1 , where \tldO hides polylogarithmic terms.
After revision:  We offer two alternative schemes that yield significantly smaller sets, and two related results, as follows: (A)Let be a sample of size ], that computes (roughly) a 1/(4d^2)-centerpoint in \tldO(d ^2 ^{-1 , where \tldO hides polylogarithmic terms.