A note on notation - [0,1] with square brackets generally refers to the closed interval between 0 and 1. X is a continuous variable, not a boolean one.
Actually, I should have been using curly brackets, as when I wrote (0,1) I meant the set with two elements, 0 and 1, which is what I had taken X to be a product of copies of, hence my obtaining 50000 as the expected Manhattan distance between any two members. I'll correct the post to make that clear. I think everything I said would still apply to the continuous case. If it doesn't, that would be better addressed with a separate comment.
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