It's unclear to me why sampling from product measures is tractable while generally randomized sampling is not. This is in the context of MC rejection sampling.
Suppose we want to sample from a hypersphere $S$ living in $\mathbb{R}^d$ and we use a hypercube as our ''bounding box'' where $S$ is inscribed in the hypercube. When randomly sampling from the hypercube and using rejection sampling, I understand that in high dimensions it becomes very likely that a given draw will be rejected due to the difference in volumes in high dimensions of our regions.
To deal with this problem, why can't/shouldn't we take a more carefully constructed bounding region, say some convex regular polytope or some convex polytope (or something even more intricate for more complicated regions $S$)? I was told that the handwave-y reason we shouldn't has to do with how sampling from product measures and their mixtures is tractable, but are exceptions to the general intractability of MC rejection sampling.