I just ask the following question:
Set invariant under group action
Furthermore, How to prove the green part
Original paper: http://arxiv.org/pdf/1403.4914v1.pdf (p.1324)
Let $$O(n)=\{X\in \mathbf{R}^{n\times n}\mid X^TX=I\}.$$
Its polar: $$O(n)^o=\{Y\in \mathbf{R}^{n\times n}\mid \langle Y,X\rangle\leq 1,\forall X\in O(n)\}.$$
My proof:
Suppose $Y=U\Sigma V^T\Rightarrow \Sigma=U^TYV$ (singular value decomposition).
I want to show $\langle U_1YV_1^T,X \rangle \leq 1$, for all $X\in O(n)$.
Let $U_1=U^T, V_1=V$, then it becomes $\langle \Sigma,X \rangle $, for all $X\in O(n)$.
Can I say $\langle \Sigma,X \rangle \leq 1$?
I have no idea how to prove or even this way is wrong. How to prove the green part?