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Sampling from a partition of a hypercube by convex polytopes.

I have a binary space partitioning (BSP) tree which recursively partitions a hypercube using linear hyperplanes. That is, a hyperplane splits the hybercube in half, creating two convex polytopes. … I'm considering partitions of 1,000-10,000 polytopes and sampling 100-1,000 points per polytope for dimensions up to 10-20 (i.e. in $\mathbb{R}^d$). …
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