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This conjecture by Semon was intriguing enough I had to test it immediately. Checks out. Anyone want to attempt a formal proof that dot(x1,x2) --> 0 as dim(X) --> inf?
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Morning thought: 2x2 diagrams work so well because a random pair of vectors in a high dimensional space are orthogonal with probability ~1 @vgr
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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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This is about the expectation value. The measure used is the Haar measure for unitary rotations (I am considering complex vectors). The inequality is due to the concavity of the square root.
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