We consider the popular k-means problem in d-dimensional Euclidean space. Recently Friggstad, Rezapour, Salavatipour [FOCS’16] and Cohen-Addad, Klein, Mathieu [FOCS’16] showed that the standard local search algorithm yields a p1`εq-approximation in time pn ̈kq Opdq , giving the first polynomialtime approximation scheme for the problem in low-dimensional Euclidean space. While local search achiev...