Question

Let \(Z_{1}, \ldots, Z_{k}\) be distinct points in \(\mathbb{R}^{n}\). Find a unique point \(X_{0}\) in \(\mathbb{R}^{n}\) at which the function
\[
Q(X)=\left\|X-Z_{1}\right\|^{2}+\cdots+\left\|X-Z_{k}\right\|^{2}
\]
achieves its minimum value by "completing the square" to obtain the identity
\[
Q(X)=k\left\|X-\frac{1}{k} \sum_{n=1}^{k} Z_{n}\right\|^{2}+\sum_{j=1}^{k}\left\|\left.Z_{j}\right|^{2}-\frac{1}{k}\right\| \sum_{j=1}^{k} Z_{j} \|^{2}
\]