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 ratio between the sketching and clustering objectives is constant over compact metric spaces . 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.