This is a companion contrast to [the earlier analogous question for unit $n$-cubes](https://mathoverflow.net/q/163294/6094),
where the answer (provided by several respondents) is $\infty$ .



> 
> What is the limit, as $n \to \infty$, of the expected distance between two
>  points chosen uniformly at random within a unit-radius hypersphere
>  in $\mathbb{R}^n$, i.e., in the unit-radius $n$-ball?
> 
> 
> 


Dividing OEIS [A093530](http://oeis.org/A093530) by [A093531](http://oeis.org/A093531) I
see that it appears to be approaching approximately $1.37$ for odd $n$, but I wonder if the
limit is actually known, either exactly or to significant precision?
I cannot quite extract an answer from the [MathWorld article](http://mathworld.wolfram.com/BallLinePicking.html)...


This certainly provides a dramatic contrast between the $n$-cube and the $n$-sphere!



Limit of distance between two random points in a unit-radius $n$-sphere
Both points will be very close to (let's pretend: on) the surface with prob almost 1. Call the first point the north pole. By concentration of measure for the sphere, a randomly chosen second point is almost guaranteed to be almost on the equator, so the limit should be $\sqrt{2}$.


