This reminds me of the well-known counterintuitive result that in high dimensions, most of the weight of a distribution is in the tails rather than towards the center. Possibly related I think.
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Ah looks like it's well-knownhttps://twitter.com/blakejriley/status/1293931550384553986 …
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That particular point is so simple it may be hard to dig up a citation (assume unit vectors uniformly distributed, rotate so that one is (1,0,0,0...), what's the average squared value of a single coordinate of the other?).
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People interested in this question might also find my dissertation interesting; it covers a bunch of related stuff. https://ansuz.sooke.bc.ca/books/thesis.pdf …
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See Blessing of Dimensionality in this link: https://en.wikipedia.org/wiki/Curse_of_dimensionality …
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