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From wikipedia the definition of diameter is the supremum of the distance function of the set. But what if there is no obvious distance function, say for the set $SO(n)$. Also how does this work when distance is just some function, what if I replace distance $d$ with $D(p_1,p_2)=2d(p_1,p_2)$, then the supremum of their differences would be bigger, but this doesn't change the diameter?

I do know that $SO(2)$ is isomorphic to $S^1$, does that mean I can say the diameter is 2?

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    @QiaochuYuan, it was a question unattached to a textbook, just "What are the intrinsic diameters of SO(2n) and SO(2n+1)?" The textbook we have been using is Frank Morgan's "Riemannian Geometry". My professor probably assumed that the choice of metric would be immediately obvious, but it is not, at least to me.2012-12-03

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