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Sampling from a partition of a hypercube by convex polytopes.

This results in a partition of the hypercube into convex polytopes, where each polytope is defined by a set of linear inequalities $Ax < b$. … I understand there are various ways of sampling from convex polytopes (e.g. rejection sampling, MCMC), but I wonder if there is any way to exploit the structure of my problem - in particular that the polytopes
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