Before revision:  This was subsequently improved to a 1/\sqrt{\varepsilon\mathcal{O}\left({\varepsilon^{4/p}}\cdot n^{1-2/p}}\right) \mathcal{O}\left({\varepsilon^{4/p}}\cdot n^{1-2/p}}\right)  the first optimal dependence on the approximation factor.
After revision:  This was subsequently improved to a 1/\sqrt{\varepsilon\mathcal{O}\left(\frac{1{\varepsilon^{4/p}}\cdot n^{1-2/p}}\right) \mathcal{O}\left({\varepsilon^{4/p}}\cdot n^{1-2/p}}\right) the first optimal dependence on the approximation factor.