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?



Smoothness of Minkowski functional is equivalent to smoothness of boundary
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](https://www.mat.univie.ac.at/%7Emichor/apbookh-ams.pdf).


