Concentration of norm of projection onto a subspace Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of $\mathbb{R}^n$ of dimension $k$ and let $P_V(x)$ be the orthogonal projection of $x$ onto $V$.
I have seen quoted in the literature that 
\begin{align}
\mathbb{P}[|\left\| P_V(x)\right\|_2 - \sqrt{k/n} | \le \epsilon] \ge 1 -2\exp(-n\epsilon^2/2). \, \, \, \, \, \, \, (1)
\end{align} However, i can still not find a concrete proof. What i do understand is that for a $1$-Lipschitz function $f:\mathbb{S}^{n-1} \rightarrow \mathbb{R}$ such as $x \mapsto |\left\| P_V(x)\right\|_2$, we have that 
\begin{align}
\mathbb{P}[|f - M_f | \le \epsilon] \ge 1 -2\exp(-n\epsilon^2/2), \, \, \, \, \, \, \, (2)
\end{align} where $M_f$ is the median of $f$. (2) mostly follows from the isoperimetric inequality on the sphere. The issue though with (1) is that $\sqrt{k/n}$ does not seem to be the median of $x \mapsto |\left\| P_V(x)\right\|_2$. Is anyone able to provide a clean argument for (1) or a self-contained reference in the literature? Many thanks.   
 A: Have you checked Artstein's Proportional concentration phenomena on the sphere? He is addressing the question of such estimates (in section $6$), although I have not been able to find this exact one.
Or Theorem 7.5 of this paper. It is a version of your estimate where the median is replaced with the average. Not exactly what you are looking for either but it feels we are getting close, maybe a combination of these methods?
A: I believe result is a version of the so-called Johnson-Lindenstrauss Lemma which was initially proved by analytic methods. The probabilistic version was proved in the paper referenced below. See here for a link.
Dasgupta and Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures and Algorithms, 22:60-65, 2002.
http://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf
This seems to be a key observation in the paper:

Hence the aim is to estimate the length of a unit vector in $R^d$ when it is projected onto
  a random $k-$dimensional subspace. However, this length has the same distribution as the
  length of a random unit vector projected down onto a fixed $k-$dimensional subspace. Here
  we take this subspace to be the space spanned by the first $k$ coordinate vectors, for
  simplicity.

