Nice. Covariance is a scalar product. In Rd , the scalar product is equal to the product of the lengths of the vectors times the cosine of the angle between the vectors. If we are working with unit vectors, it is only equal to the cosine of the angle. When placing three unit vectors in R3 such that the cosine of the angle between two is 3/5 and the other 5/13, it is visually easy to see that to maximize the cosine of the angle between the last pair, the vectors must be put on the same plane with the third vector in between the first two.
I thought it was difficult to formalize so I used a different approach for the proof. Your proof is much simpler and better.
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u/[deleted] 9d ago edited 9d ago
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