[coldcache]

> The advantage of this approach is that it generalizes efficiently to any number of dimensions.

I am unsure about whether this is true. The ratio of a ball’s volume to its enclosing hypercube’s volume should decrease to 0 as dimensionality increases. Thus, the approach should actually generalize very poorly.

[alberto_balsam]

Note that the author is not referring to the accept-reject method here

[coldcache]

Ah I misread this, thanks

[scythe]

Let S = {S_i} be any set of cubes that covers a d-sphere. Choose a point in a cube and an integer i in [0, |S|). Now you have a random point in S. With a judicious choice of S you obtain a uniformly random point in the unit sphere with high probability.