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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.

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    $\begingroup$ 1) The boundary of $\partial K$ is $C^2$ if and only if $g_K$ is $C^2$, but that's almost a tautology. 2) There is a pointwise relationship between the second fundamental form at a point on $\partial K$ with the second fundamental form of $\partial K^*$ at the point in the direction the unit normal of $\partial K$. An affine description of this can be found in deaneyang.com/papers/affine_survey.pdf (see equation (10), but it can be translated into a Euclidean statement with some effort. 3) I can't help with this unless you say more about what you're looking for. $\endgroup$ – Deane Yang Nov 19 '15 at 19:17
  • $\begingroup$ Thanks Deane. This almost solved my problem. A few questions though, if $\partial K$ is $C^2$ on some region with positive curvature, can we say that $\partial K^\circ $ is also $C^2$ on some "dual" region? Also, what is the answer to question (1) when we replace the gauge function by the support function? $\endgroup$ – Mohammad Safdari Nov 20 '15 at 15:45
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    $\begingroup$ The answer to the first question is yes. The dual region is the intersection of $\partial K^*$ with the cone spanned by the image of the Gauss map. The Gauss curvature on a region in $\partial K$ is positive and continuous $\iff$ the Gauss curvature is positive and continuous on the dual region $\iff$ the second fundamental form is continuous and positive definite $\iff$ the second fundamental form on the dual region is continuous and positive definite. The latter two are equivalent to the boundary (or dual boundary) being $C^2$. $\endgroup$ – Deane Yang Nov 23 '15 at 19:37
  • $\begingroup$ Since the support function is the gauge function of the polar, it is $C^2$ if and only if $\partial K^*$ is $C^2$. $\endgroup$ – Deane Yang Nov 23 '15 at 19:38
  • $\begingroup$ @DeaneYang, Thanks a lot. If you would like you can post your comments as an answer, and I'll accept it. $\endgroup$ – Mohammad Safdari Nov 24 '15 at 12:44
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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.

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  • $\begingroup$ I just want to add a comment about item (1). The mere fact that the graph of a function is a $C^2$ manifold does not imply that the function is $C^2$. Even if we think of $\partial K$ as a level set of $g_K$, we cannot deduce that $g_K$ is $C^2$, as can be seen by simple examples like $f(x,y)=\phi (x^2 + y^2)$. But thinking in spherical coordinates, we can easily construct $g_K$ from the $C^2$ function from the unit sphere whose graph is $\partial K$. $\endgroup$ – Mohammad Safdari Nov 27 '15 at 16:50
  • $\begingroup$ In the above comment, by the graph, I actually meant image. As it is obvious that the regularity of a function between manifolds $M$ and $M\times N$ is the same as the regularity of its graph. This is actually what happens here. Also to elaborate the last part, if $\partial K$ is locally given by an equation $r=f(u)$ for $u\in S^n$, then it is easy to see that $g_{K}(x)=|x|/f(x/|x|)$. $\endgroup$ – Mohammad Safdari Nov 28 '15 at 14:54

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