I am considering a point R inside an sphere of radius 1 in dimension d. The point is at a distance x from the center of the sphere, and the axis 0x is thus defined.
The distance between the point and the sphere surface is $L(x,\mathbf{\omega})$, where $\mathbf{\omega}$ is the angle between R and the point considered on the sphere. In 2D, $\mathbf{\omega}$ is an angle $\theta$, in 3D, it is $\theta,\phi$, etc...
I am interested in the mean distance (to a power $p$) between R and the sphere surface, projected on the axis Ox. The average is done over all possible angles (rather than over all points of the sphere surface). Physically, this value can be seen as the net force $f$ on R along Ox, when for all angles, the point measures $L$ and gets pulled to the surface with a force $L^p$.
For exemple, in d=1 :
$f^1_p (x)=(1+x)^p-(1-x)^p$
In d=2 :
$f^2_p (x)= \int_0^{2 \pi} d\theta \sin(\theta)L(x,\theta)^p$
with $L(x,\theta)=\sqrt{1-x^2 + x^2 \sin^2(\theta)}$
Interestingly enough, I found numerically that this measure has very funky properties. Namely, $f_p^d$ is linear if p=1 or p=d+1, and is sublinear in between.
My goal was to prove this linearity for p=d+1 recursively, but I am now stuck. Any chance this is a known problem, or that there is a simple solution I overlooked?
Note : If I'm correct, one can show that :
$f_p^d = 2 \int_0^1 \frac{u^{d-1}}{\sqrt{1-u^2}} f_p^{d-1} \left( \frac{x}{R_1(x,u)} \right) \left(u R_1(x,u) \right)^p du \qquad $ With $R_1(x,u)= \sqrt{ \frac{R_0^2 - x^2}{u^2} + x^2 } $
Edit : numerically, it seems that the mean p-distance $<l_d^p>$from R to the sphere, averaged over all angles $\omega$ is a constant if $p=d$. I'll call this constant $C_d$.
Let us take that as an assumtion to compute $f_{d+1}^d$ :
$f_{d+1}^d = \int_\Omega d\omega \mathbf{u}_x \mathbf{.} \mathbf{u}_\omega L(x,\omega)^{d+1} $
I thought I could integrate by parts, using $f_1^d = \int_\Omega d\omega \mathbf{u}_x \mathbf{.} \mathbf{u}_\omega L(x,\omega)^1 $ and using $f_{1}(x)=x$. However, this does not really achieve anything.
I wrote the solution in a PDF eventually. It's not very complicated algebra but it is a bit long.
http://biophysics.fr/wp-content/uploads/2018/06/derivations-1.pdf