Unfortunately, there is an error in Lemma 13 of this paper (https://arxiv.org/pdf/1408.4045.pdf) that affects the result about the K-medians LP, you can find more details in https://arxiv.org/abs/2109.02547 (in a nutshell, rotational symmetry of the distributions is not enough to give a required symmetry in part of the argument). Many thanks to the authors of arXiv:2109.02547, Del Pia and Ma, for finding this error. The results about k-means are independent of this Lemma and are, to the best of my knowledge, correct.