In this 1-page article, the author claims that
the “expected” magnitude of the projection of a vector onto a random unit vector is proportional to its length [...] and it is nonzero.
However, I have emailed the author by asking how to calculate that expected value exactly, I got the response
[...]taking a surface integral over the n-dimensional sphere of the given quantity shown in the bracket.
So, if the proposed definition of the expected value function $E$, the integral has to be zero, because, we are evaluating a constant vector over S^n, and, for example, for $n=2$, we can employ the divergence theorem. Since the divergence of the constant vector is zero, the surface integral has to be zero. I mean even without the divergence theorem, for a constant vector $v$ and any unit vector $u$, $v \cdot u$ = - ( v \cdot (-u) ), so even intuitively I would expect this "expected" value to be zero.
So is there anything that I'm missing or just the author is somehow wrong in his claim ?
I have sent my solution to the author, but got no response.