Before revision:  Specifically we show a polynomial time construction for a coreset of size O( \sqrt{d\log (1/\epsilon) / \epsilon ), and we show a near-matching lower bound of size \Omega(  d/ \epsilon).
After revision:  Specifically we show a polynomial time construction for a coreset of size O( \sqrt{d\log (1/\epsilon) / \epsilon ), and we show a near-matching lower bound of size \Omega( d/ \varepsilon, 1/\varepsilon^2\}). When d\geq 1/\varepsilon^2, it is known that the size of coreset can be O(1/\varepsilon^2).