Before revision:  The former result is the first known coreset construction for the fair clustering problem with the k-median objective, and the latter result removes the dependence on the size of the full dataset as in Schmidt et al. and generalizes it to multiple, non-disjoint types.
After revision:  The former result is the first known coreset construction for the fair clustering problem with the k-median objective, and the latter result removes the dependence on the size of the full dataset as in 39 and generalizes it to multiple, non-disjoint types.