Before revision:  The number of stabbing queries performed is O( d^2\log ^{-1 \tldO(d ^9 ^{-1 is the first weak -net related construction where all constants/bounds are polynomial in the dimension. (B) If one is allowed to expand the convex range before checking if it intersects the sample, then a sample of size ( ^{-(d+1)/2), from the original point set, is sufficient to form a net.
After revision:  The number of stabbing queries performed is O( d^2\log ^{-1 \tldO(d ^9 ^{-1 is the first weak  ( ^{-(d+1)/2), from the original point set, is sufficient to form a net.