How do I feasibly generate a random sample from an $n$-dimensional $\ell\_p$ ball? Specifically, I'm interested in $p=1$ and large $n$. I'm looking for descriptions analogous to the statement for $p=2$: Take $n$ standard gaussian random variables and normalize.



How to generate random points in $\ell_p$ balls?
For arbitrary p, [this paper](http://arxiv.org/abs/math/0503650) does exactly what you want. Specifically, pick $X\_1,\ldots,X\_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector
$$\frac{(X\_1,\ldots,X\_n)}{(Y+\sum |X\_i|^p)^{1/p}}$$
is uniformly distributed in the unit ball of $\ell\_p^n$.


The paper also shows how to generate certain other distributions on the $\ell\_p^n$ ball by modifying the distribution of $Y$.


