Computing the average of $\prod_i (1-\frac{|x_i|}{L})$ on the surface of the unit $n$-sphere

I'm trying to compute the expected value of $\prod_i (1-\frac{|x_i|}{L})$ on the surface of a $n$-dimensional sphere. A first step could be to integrate only on the first quadrant to take out the absolute value (since the function is symmetric).

The intuition behind it (if I'm not wrong) is that for $L>1$ that product is the probability that, if we put edge-parallel planes in every dimension separated by distance $L$, a point in the surface of the unit sphere will end up in the same hypercube as the origin.

For 2 dimensions it's easy because we can pass to polar coordinates:

$$\frac{2}{\pi}\int_{\theta=0}^{\pi/2} (1-\frac{\cos \theta}{L})(1-\frac{\sin \theta}{L}) = 1- \frac{4L-1}{\pi L^2}$$

• Did you try looking at $\prod_i \left(1-\frac{\mid x\mid}{L} \right)$ as $\sum_i \log\left(1-\frac{\mid x\mid}{L} \right)$? – Guilherme Thompson Nov 28 '16 at 17:07
• And, on the surface of the n-hypersphere, one could say that $\mid x\mid = r$, no? – Guilherme Thompson Nov 28 '16 at 17:10