For the uniform sampling from general convex polytopes, see e.g. [Answer 1][1], [Answer 2][2], other answers to this [Question][3], and further references given there. Also, for instance, [Fast MCMC Sampling Algorithms on Polytopes][4]. 


  [1]: https://mathoverflow.net/a/162327/36721
  [2]: https://mathoverflow.net/a/9858/36721
  [3]: https://mathoverflow.net/q/9854/36721
  [4]: https://jmlr.org/papers/volume19/18-158/18-158.pdf