Let $\mathcal{K}$ be a compact, convex set in $M_n(\mathbb{C})$ and $X\in M_n(\mathbb{C})$ s.t. $X\notin\mathcal{K}$. Then show that there exist $\theta\in [0,2\pi)$ and $\mu\in\mathbb{R}$ s.t. $$e^{i\theta}\mathcal{K}\subseteq\{Y\in M_n(\mathbb{C}): Y+Y^*\leq \mu\}$$ and $$ e^{i\theta}X+e^{-i\theta}X^*>\mu.$$ where $M_n(\mathbb{C})$ is the collection of $n\times n$ matrices with complex entries.
Comments: I can see this whenever $n=1$ i.e. whenever $\mathcal{K}$ is a compact, convex set in $\mathbb{C}$. But I am not getting any clue to prove the general statement.
Any comment is highly appreciated. Thanks in advance.
This is not true. Let $$ \mathcal K=\operatorname{conv}\left\{\begin{bmatrix} -1&0\\0&0\end{bmatrix},\begin{bmatrix} 1&0\\0&0\end{bmatrix}\right\}=\left\{\begin{bmatrix} 1-2t&0\\0&0\end{bmatrix}:\ t\in[0,1]\right\},\ \ \ \ \ X=I_2. $$ The first requirement is that $$ \mu\geq e^{i\theta}\begin{bmatrix} 1-2t&0\\0&0\end{bmatrix}+e^{-i\theta}\begin{bmatrix} 1-2t&0\\0&0\end{bmatrix}=\begin{bmatrix} 2(1-2t)\cos\theta&0\\0&0\end{bmatrix}. $$ Taking $t=0$, a necessary condition on $\theta$ is that $$\tag1 \cos\theta\leq\frac\mu2. $$ The second requirement is that $$ \mu<e^{i\theta} I_2+e^{-i\theta}I_2=2\cos\theta I_2, $$ so $$\tag2 \cos\theta>\frac\mu2. $$ Conditions $(1)$ and $(2)$ together are impossible.