 Solution

Write $n_k=|C_k|$ and suppose the cluster is nonempty. Expanding the Euclidean norm through its inner product givesSum over $x\in C_k$. The final term does not depend on $x$, so its sum is $n_k\Vert z-{\bar{x}}_k{\Vert }^2$, which proves the requested identity. This expansion is the starting point for the centroid sum-of-squares identity.