There are some pretty simple methods to do uniform sampling on the surface of high-dimensional spheres or cubes. Are there any methods that sample uniformly on the surface of a high-dimensional polytope?

Given a high-dimensional polytope like this:

$\| A \mathbf{x} \|_{\infty} = 1$

where $A$ is a given $m \times n$ matrix and $\mathbf{x}$ is a $n \times 1$ vector.

We want to sample a set of $\{ s_1, s_2, \dots, s_n \}$ that lie uniformly on the surface of this polytope.

I'm looking at some advanced Monte Carlo Methods, but it's not that easy.

To make the problem simpler, can we find some methods that can do infinite (non-uniform) sampling on the surface of the polytope?