Ball point picking is the selection of points randomly placed inside a ball. 
random points can be picked in a unit
ball in the Wolfram Language using
the function RandomPoint[Ball[], n].
Pick variates 
,
..., 
independently from a normal distribution with
mean 0 and variance 
,
and variate 
independently from an exponential distribution
with parameter 
.
Then the distribution of points
 |
(1)
|
is uniform over the 
-dimensional
unit ball. Note however that in practice, computation
of 
points inside a 
-dimensional
unit ball may still be slower using this technique than
computing additional points (e.g., by a factor of 
for 
and 
for 
) inside an 
-cube of edge length 2 and discarding the ones with norm
greater than 1.
This result is a special case of a beautiful general result described by Barthe et al. (2005) which may be stated as follows. For 
and a sequence of real numbers 
, define the 
-norm as
 |
(2)
|
The space of all infinite sequences with 
is then denoted 
, and the space 
equipped with quasi-norm 
is denoted 
. Finally, the unit ball of
is defined as 
.
Now, pick 
,
..., 
independently with probability density
function given by
 |
(3)
|
where 
is the gamma function, and 
an independent random variate from an exponential
distribution with mean 1. Then the random vector
 |
(4)
|
is uniformly distributed over the unit ball of 
(Barthe et al. 2005).
-Ball." Ann. Probab. 33, 480-513, 2005.