Before revision:  We show that the gap is constant for any compact metric space . 2. We extend the above results to obtain constant gaps for the case of mm-spaces, where \rho is the p-Gromov-Wasserstein distance and the  clustering objective involves minimizing various notions of the \ell _p-diameters of the clusters.
After revision:  We show that the gap is constant for any compact metric space . 2. We extend the above results to obtain constant gaps for the case of mm-spaces, where \rho is the p-Gromov-Wasserstein distance and the setting of metric measure spaces where we prove that the ratio of sketching to clustering objectives is bounded both above and below by some universal constants. In this setting, the clustering objective involves minimizing various notions of the \ell _p-diameters of the clusters.