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.
After revision:  The number of stabbing queries performed is O( d^2\log ^{-1 \tldO(d ^9 ^7). While the improvements are (arguably) mild, it is the first weak -net related construction where all constants/bounds are polynomial in the dimension.