Take two points, $p_0$ and $p_k$, in $n$-dimensional Euclidean space, where $d(p_0,p_k)$ is the distance between the points. Now, draw an $n$-sphere of radius $r$ centered on $p_0$ and uniformly select a new point, $p_1$, in the volume of the sphere, and a new point $p_2$ on the surface of the sphere.

Let $h_1 = d(p_0,p_k) - d(p_1,p_k)$, and $h_2 = d(p_0,p_k) - d(p_2,p_k)$, represent the difference in the distance from $p_0$ to $p_k$ if $p_0$ is moved to $p_1$ or $p_2$, respectively. To clarify, if we blow up another $n$-sphere, $S$, about $p_k$ until $p_0$ "touches" its contour, $h_1$ / $h_2$ will be positive if $p_1$ / $p_2$ is inside $S$, zero if $p_1$ / $p_2$ are on the contour of $S$, and negative if $p_1$ / $p_2$ fall outside $S$.

What probability distribution and expectation do we have for $h_1$ and $h_2$?

Update [2/15/2014] :: Thanks to Bjørn Kjos-Hanssen's efforts, we have a nice exact expression for the $n = 2$ case for $h_1$ (though it's likely one will have to resort to numerical integration to compute the expectation --- $\int_{1-r}^{1+r} \space R \times f(R) \space dR$). At this point, I think its probably wise to restrict the focus or scope of this question to the $n = 2$ case. Can a PDF for $h_2$ (where we select points along the contour of the circle) be derived in a similar manner?