I originally had this question while I was reading Michael Hardy's high-level proof here: https://math.stackexchange.com/q/56744
He states that "the mapping $(X_1,\ldots, X_n) \mapsto (X_1 - \overline{X}, \ldots, X_n - \overline{X})$ is a projection onto a space of dimension $n-1$. Notice also that its expected value is $0$. Then remember that the probability distribution of the vector $(X_1,\ldots, X_n)$ is spherically symmetric. Therefore so is the distribution of its projection onto a space of dimension one less."
I tried constructing a rigorous proof of why the statement above about spherical symmetry is true, but my linear algebra/geometry chops are not good enough to get me there. Namely, I assume one should work with a projection matrix, but I'm not sure how to tie in the idea of spherical symmetry.
Can someone help me in thinking about how to prove this?
Any given rotation in an $(n-1)$-dimensional subspace can be lifted to a rotation in the $n$-dimensional space by leaving the coordinate orthogonal to it invariant. Rotating the rotationally symmetric distribution in the $n$-dimensional space by the lifted rotation rotates the projected distribution in the $(n-1)$-dimensional space by the given rotation. Since the symmetric distribution doesn’t change, neither does the projected one.