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](http://rads.stackoverflow.com/amzn/click/0521352207).


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.



Questions about the regularity of the "norm" associated to a convex set
Although the answers to these regularity questions are not addressed explicitly in the paper cited in the comments to the question, they all follow from the formulas derived in that paper.


1) The answer is yes. The boundary $\partial K$ is the radial graph of its gauge function over the unit sphere. A hypersurface is defined to be $C^2$ if it is locally the graph of a $C^2$ function, which in turn holds if and only if the gauge function is $C^2$.


2) If $\partial K$ has strictly positive Gauss curvature, then at each point $x \in \partial K$ there is a formula for the second fundamental form of $\partial K^\*$ at the corresponding "dual" point (given by the Gauss map) in terms of the second fundamental form and gauge function of $\partial K$ at $x$. Taking the determinant gives an explicit formula for the corresponding Gauss curvatures.


An affine invariant version of this formula is given in the paper cited in the comments to the question. The Euclidean version of the formula can be derived with some effort, from the affine one.


Additional questions asked in comments:


(Aside: It's easier to work with the square of the gauge function.)


a) There is an explicit formula for the Hessian of the squared gauge function in terms of the Hessian of the squared polar gauge function. From this, it follows that $\partial K$ is $C^2$ and the Gauss curvature positive at a point $\iff$ the Hessian of the squared gauge function is positive definite and continuous $\iff$ the Hessian of the squared polar gauge function (i.e., the squared support function of $K$) is positive definite and continuous $\iff$ $\partial K^\*$ is continuous and the Gauss curvature positive at the dual point on $\partial K^\*$.


b) Since the support function of $K$ is the gauge function of $K^\*$, the support function of $K$ at a dual point is $C^2$ if and only if $\partial K^\*$ is $C^2$ at that dual point.


ADDED: One more comment regarding the use of gauge functions (i.e., convex functions homogeneous of degree 1) in PDE's and the calculus of variations. Often, the Euclidean norm plays no role in the setup of the problem (i.e., the definition of the constraints and objective functional), even though they are needed for defining the functional norms when proving estimates. In that situation it is easiest (at least for me) to work out affine invariant formulas, where the Euclidean norm and inner product are never used. When doing this, it is important to keep track of what lies in the vector space that contains the convex body and what lies in the dual vector space. All of this is discussed pretty carefully in the paper cited. I would add that when I was learning all of this, I found Rockafellar's book *Convex Analysis* to be extremely helpful.


