Before revision:  Given a convex body , via a separation oracle, the algorithm performs a small sequence of (oracle) stabbing queries (computed from ) -- if none of the query points hits , then contains less than an -fraction of the input points. 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:  Given a convex body , via a separation oracle, the algorithm performs a small sequence of (oracle) stabbing queries (computed from ) -- if none of the query points hits , then contains less than an  ^{-1 \tldO(d ^9 ^{-1 is the first weak -net related construction where all constants/bounds are polynomial in the dimension.