Let $B^{n} \_p= ${$ (x\_1, \dots, x\_n) : |x\_1|^p + \dots |x\_n|^p = 1 $} be the unit ball in $\mathbb{R}^n$ in the $\ell^p$ norm.


If $X\_1,\dots,X\_n$ are iid $\exp(1)$ -distributed random variables, then $(X\_1/D,\dots,X\_n/D)$, where $D =X\_1+ \dots + X\_n $ is uniformly distributed in $B^{n}\_1$.


If $X\_1,\dots,X\_n$ are iid normally distributed with mean 0, then $(X\_1/D,\dots,X\_n/D)$, where $D = (X\_1^2+\dots+X\_n^2)^{1/2}$, is uniformly distributed in $B^{n}\_2$.


Is there a choice of $X\_1,\dots , X\_n$ iid such that 
$ ( X\_1 / D, \dots, X\_n/D)$, where $D = (|X\_1|^p + \dots + |X\_n|^p)^{1/p} $ is uniformly distributed in $B^{n} \_p$ for arbitrary $p$?


I would be happy with any sensible common generalization of the two statements above. I have no particular reason to believe there is such a generalization - I'm just hoping that two so similar and neat examples have similarly nice generalizations.



Sampling uniformly from a sphere
The result you want, I think, is in  [Stationarity, Isotropy and Sphericity in $l\_p^\*$](http://www.springerlink.com/content/r2771gx9j2g40132/). It is behind a pay-wall, but the form of the distribution is stated in the abstract.


