In Lloyd's algorithm, for every cluster C, we choose the cluster center to be the centroid μ where μ = 1 P xi. However, there may be other ways of choosing the cluster centers |C| i∈C for k-means, for a cluster C. Let Z(C, z) = Pi∈C ∥xi − z∥2 be the "goodness" of cluster C with center z. Prove that μ is the optimal center for the cluster C, that is