In a paper by Alan Edelman, "The geometry of algorithms with orthogonality constraints" (page 35 ), there are several definitions to the notation "distance" between vector spaces on the Grassmanian. Let Y1 and Y2 be orthonormal matrices, I need an answer of "true" or "false" if the distance
$$d_{cf}(Y_1,Y_2)= \min_{O_1 ,O_2 \in O_p}\|Y_1\cdot O_1-Y_2\cdot O_2\|_{\text{Frobenius}}$$
defines a metric. Does it satisfy the triangle inequality?
Although I succeded to prove that
$$d_cf(Y_1 , Y_2)\leq \sqrt{2}(d_cf(Y_1 , Y_3)+d_cf(Y_3 , Y_2))$$
I still think the answer is "no", because I know that
$$d_cf(Y_1 , Y_2) = \|Y_1\cdot U- Y_2 \cdot V\|_{\text{Frobenius}}$$
where $U(\textbf{cos($\Theta$))}V^T$ is the singular value decomposition of $Y_1^{T}Y_2 $ and $ \Theta$ is the diagonal matrix of principal angles.