Before revision:  When d>1 is constant, our construction gives a coreset of size O\left(1{\varepsilon} \sqrt{\log\log\frac{1{\varepsilon}}} \right) as opposed to the best-known result of O\left(1{\varepsilon}\log\frac{1{\varepsilon}}\right).
After revision:  When d>1 is constant, our construction gives a coreset of size O\left(1{\varepsilon} {\varepsilon}}} \right) as opposed to the best-known result of O\left(1{\varepsilon}\log\frac{1{\varepsilon}}\right).