Let $K$ be a convex compact set in $\mathbb{R}^2$ with $0 \in \overset{\circ}{K}$. The Minkowski functional associated to K is: \begin{align*} \varphi_K(x):= \inf \left\{t>0 \; : \; tx \in K \right\} \end{align*}
Is there some results linking the regularity of $\varphi_K$ to the regularity of $\partial K$? For example, if $\partial K$ is $C^2$, can I expect $\varphi_K$ to be $C^2$? I would gladely look into any reference.
