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Let's say we have two $d$ dimensional subspaces with principal angles $\theta_1, \dots, \theta_d$. Then, if $U,V$ are the orthonormal bases for these two subspaces, the singular values of $U'V$ are $\cos\theta_1,\dots,\cos\theta_d$. Now we define $$D(U,V) = \sqrt{\det(I-U'VV'U)}= \prod_{i=1}^d \sin\theta_i$$ My question is: is the above defined $D(U,V)$ a semi-metric? Essentially, I am asking, does the triangle inequality hold here, i.e., $D(U,W)\leq D(U,V)+D(V,W)$?

There is a similar post: Does the product of principal sines between subspaces satisfy the triangle inequalilty?

But no explicit answer was given there.

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  • $\begingroup$ Say $U'U=I_d$ and $V'V=I_d$, then unless I'm completely overseeing something obvious, it is pretty easy to find counterexamples to the triangle inequality. $\endgroup$
    – Suvrit
    Nov 18, 2014 at 19:10
  • $\begingroup$ Thanks! But do you have specific counterexamples when the subspaces spanned by U,W and V are pairwisely disjoint? @Suvrit $\endgroup$ Nov 19, 2014 at 20:27

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