Generalization of parallel axis theorem

Question

The parallel axis theorem is a well known result. Is the result still true when we replace a parallel axis with arbitrary affine subspaces? That is, prove that if $\sum_{i=1}^nx_i=0$, then the sum of the squared distances from this set of points (i.e., $\lbrace x_i\rbrace_{i=1}^n$) to an affine subspace is the sum of the squared distances to the subspace plus $n$ times the squared distance from the affine space to the origin.

Answer 1

Consider the affine subspace $$ x_0+S=\{x_0+z;z\in H\},$$ where $S$ is a linear subspace of the Hilbert space $H$. Let $\Pi$ be the projection operator onto $S$. Then, the sum of the square distances from the set of points $x_i$ to $x_0+S$ is \begin{align} \sum_{i=1}^n|x_i-(x_0+\Pi(x_i-x_0))|^2&= \sum_{i=1}^n |x_i|^2+|x_0|^2-2\langle x_i,x_0\rangle - |\Pi x_i|^2-|\Pi x_0|^2-2\langle \Pi x_i,\Pi x_0\rangle \\ &=\sum_{i=1}^n |x_i|^2+|x_0|^2 - |\Pi x_i|^2-|\Pi x_0|^2\quad\text{; because $\Pi$ is linear}\\ &=\sum_{i=1}^n |x_i-\Pi x_i|^2+|x_0-\Pi x_0|^2, \end{align} where the last equality comes from $\langle z-\Pi z,\Pi z\rangle=0$, and the result follows.