I have a question about Minkowski functional.
We start with a compact, convex set $C$ containing $0$ in its interior. We define the Minkowski functional of $C$ to be the function $\mu: R^n \to [0,\infty)$ given by the formula \begin{eqnarray} \mu _C(x) &:=& \min\{\lambda ; \lambda^{-1}x \in C\}\\ \mu_C (0) &: =& 0. \end{eqnarray}
Assume now that the boundary $\Sigma$ of $C$ is a $C^2$ hypersurface. Then $\mu_C$ is $C^2$ at every point $x \neq 0$ in $R^n$.
Now my question, how can I check that \begin{equation} \mu_C'(x) = x\|x\|^{-1}, \;\; \text{for}\;\; x\neq 0. \end{equation}
I did some research and found that when $C$ is symmetric, $\mu$ define a norm at the space. But in my case, $C$ is not symmetric.
Another thing that may help is that: Let $x \in R^n$, $x \neq 0$, so the segment 0x intersect the boundary $\Sigma$ in exactly one point $x'$. So \begin{equation} \mu_C(x) = \frac{d(0,x)}{d(0,x')} = \frac{\|x\|}{\|x'\|}. \end{equation} But I don't know how to differentiate $x'$.
Thank you for the help.