Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$, $$ f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\}, $$ is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?
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Note that since you are on a finite dimensional space, the Minkowski functional yields a norm. For this norm the boundary of the convex body is the unit sphere. Now the differentiability condition for the boundary is equivalent to asking the unit sphere to be a $C^1$-submanifold.
Now on a Banach space a norm is (off 0) $k$- times continuously differentiable If and only If its unit sphere is a $C^k$-submanifold. A reference for this result is for example Theorem 13.14 in Kriegl and Michors "The convenient setting of global Analysis". Available for free here 1.
answered Jan 7, 2022 at 17:32
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$\begingroup$ It's only a norm when the domain is balanced I think. $\endgroup$ Commented Apr 14, 2023 at 12:52
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$\begingroup$ That is correct and was not mentioned in my answer. One should recheck the proof but from the top of my head it does not use the norm property in the step in an integral way. $\endgroup$ Commented Apr 16, 2023 at 8:17
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$\begingroup$ Yes, that seems reasonable to me. $\endgroup$ Commented Apr 16, 2023 at 15:21
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$\begingroup$ Maybe a question concerning this matter: is the Minkowski functional $f_C(x)$ always differentiable on the set of points $\{x\in\mathbb{R}^n: f_c(x)>1\}$, i.e. in the points that are outside the set $C$? Intuitively this should be true, since a similar result also holds true for the distance function to some convex set $C$. Would be great if somebody has a reference! $\endgroup$ Commented Sep 4 at 6:27