Given points $x_1, \ldots, x_n$ in the Euclidean space and $K \in \mathbb N$, I'm interested in the following objective. Partition the points into $K$ clusters $C_1, \ldots, C_K$ so that: $$\max_{i \in [K]} \frac{1}{|C_i|}\sum_{j \in C_i} \|x_j - \mu(C_i))\|^2$$ is minimized, where $\mu(C_i) = \frac{1}{|C_i|} \sum_{j \in C_i} x_j$. So it's similar to k-means, but it has $\max$ instead of sum, and the term for each cluster is normalized. Intuitively, I want to minimize $\max_i Var[C_i]$.
Does this problem have a name? Is there a known approximation for this problem?
