Proof of matrix inequality involving trace and max-operator Let $C\in\mathbb R^{N\times N}$ be a real positive-semidefinite matrix, $I\in\mathbb R^{N\times N}$ the identity matrix and $W,V\in\mathbb R^{N\times N}$ two real-valued random matrices. Is it true that
$$tr\left[(C-I)WCV^T\right]\leq max\left\{tr\left[(C-I)WCW^T\right],tr\left[(C-I)VCV^T\right]\right\}$$
holds for all $W, V$? My simulations suggests the inequality is true, but I have no idea how to proof it.
Any comments and ideas are greatly appreciated! Thanks!

The inequality does not hold in general. 
Let 
$$
C=1/2\,I=\begin{bmatrix}1/2&0\\0&1/2\end{bmatrix},\ \ W=\begin{bmatrix}0&0\\1&0\end{bmatrix},\ \ V=W^T=\begin{bmatrix}0&1\\0&0\end{bmatrix}.
$$
Then
$$
\mbox{tr}((C-I)WCV^T)=-\frac14\,\mbox{tr}(WV^T)=-\frac14\mbox{tr}(W^2)=0.
$$
And
$$
\mbox{tr}((C-I)WCW^T)=-\frac14\,\mbox{tr}(WW^T)=-\frac14,
$$$$
\mbox{tr}((C-I)VCV^T)=-\frac14\,\mbox{tr}(VV^T)=-\frac14,
$$
