Suppose you have a strictly convex set $\Omega \subset \mathbb{R}^n$, $n \geq 0$, bounded, with non-empty interior. Without loss of generality I will assume $0 \in \text{Int} \Omega$. Thus there exists a clear homeomorphism $x \to \frac{x}{\|x\|}$ between the boundary of $\Omega$ and the sphere.
Seems to me that this map is actually a $C^1$-diffeomorphism, is it true without further assumption ? What if we also assume the boundary of $\Omega$ to be $C^1$ ?