Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all the properties of a norm on $\mathbb{R}^n$ except for $g_K(-x)=g_K(x)$, and it is a norm when $K=-K$. We can think of $K$ as the unit ball of the "norm" $g_K$.

It is well known that when $\partial K$ is $C^2$ and its principal curvatures are all positive everywhere, then $g_K$ is $C^2$. It is easy to see that this condition is not necessary and we can allow the curvatures to be zero at least on a finite set, for example by looking at the $p$-norms for $p>2$.

Another notion here is that of the polar of $K$ which is defined as $K^\circ :=\{x\mid x\cdot y\le 1 \forall y\in K\}$. The "norm" $g_{K^\circ}$ is a "dual" to the "norm" $g_K$, much like $p,q$-norms when $1/p+1/q=1$. It is also well known that when $\partial K$ is $C^2$ and its principal curvatures are all positive everywhere, then $\partial K^\circ$ is $C^2$ and its principal curvatures are all positive everywhere too.

A good reference to all of these is the book by Rolf Schneider.

My questions are:

**1) Are there any necessary and sufficient conditions on $\partial K$ that imply $g_K$ is $C^2$?**

**2) Is there any relation between the curvature of $\partial K$ and the curvature of $\partial K^\circ$ supposing both of them are $C^2$ except possibly at finitely many points, at least when $n=2$?**

**3) Any reference that has more on the regularity of these "norms" than the above book, is also appreciated.**

Edit: These "norms" appear naturally in studying gradient constraints in PDE and calculus of variations, and their regularity has implications about the regularity of the solutions of the aforementioned problems.