Let $d \geq 3$ and suppose that $K \subset \mathbb{R}^{d}$ is a convex body (compact, convex, non-empty interior). Is the following true?

The boundary $\partial K$ is a $C^1$-manifold if and only if for each projection $\pi:K\rightarrow H$ to a hyperplane $H$ has the property that $\partial \pi(K)$ is a $C^1$-manifold.

I suspect the answer is true. I am most interested in the case $d=3$.

If it is true, does it hold true for $C^1$ replaced with $C^k$, $k \in \mathbb{N} \cup \{\infty\}$?

I have no familiarity with this field so I have no idea about the difficulty of the question. I would appreciate any references.