What is the subdifferential of the following norm at the origin
\begin{align} \lVert\eta\rVert_{\mathbf{K}}=(\eta^{\textrm{T}}\mathbf{K}\eta)^{\frac{1}{2}} \end{align} where $\mathbf{K}$ is a positive definite matrix
Question: What is the subdifferential at the origin?
On
This is just a partial answer: We want to find $\mu \in \mathbb{R}^n$ such that $$\langle K\eta,\eta\rangle\geq\langle\mu, \eta\rangle^2,\ \forall\ \eta\in\mathbb{R}^n\tag{1}$$
Let $$f(\eta)=\frac{\langle K\eta,\eta\rangle}{\langle\mu, \eta\rangle^2}$$
Note that $f(\lambda\eta)=f(\eta)$, hence the problem consists in find $\mu$ in such a way that the minimum value of $f$ in $S=\{\eta\in\mathbb{R}^n:\ |\eta|=1\}$ is bigger than or equal to $1$. Define $A$ by $$A=\{\mu\in\mathbb{R}^n\ :\ \langle\mu,\eta\rangle^2\leq\lambda,\ \forall\eta\in S \}$$
where $\lambda$ is the least eigenvalue of $K$. If $\mu \in A$, we have that $\mu$ satisies $(1)$, now the question is: Is the any element outside $A$ which satisfies $(1)$?
\begin{align} \{\frac{\alpha\mathbf{K}\eta}{\lVert\eta\rVert_\mathbf{K}}\mid0\le\alpha\le1,\eta\ne\mathbf{0}\} \end{align}