Random Points on Convex Sets (Superellipsoids)

Hi.

I am trying to find a way to draw random points from convex sets in arbitrary dimensions. In particular, the sets are described by a continuous, differentiable, and quasiconvex function such that the sets are given by . So the problem is finding a random point in uniformly distributed on , i.e. each point in is drawn with the same probability.

I reckon that the above problem is too general to admit an easy solution. But for my purposes, it would already be sufficient to find a way to randomly draw a point from sets given by , with (apparently, these sets are the boundaries of so called "superellipsoids").

Now, I do know how to draw random points uniformly distributed on spheres. I also know how to use that knowledge to draw random points uniformly distributed on ellipsoids, using a (rather inefficient) acception-rejection algorithm. But I am clueless about how to tackle the more general (second) problem described above.

Any help would be appreciated.

Cheers.