In the article of Massari presented here there is a trace inequality which is said to be true for domains which satisfy the interior sphere condition:

There exists $\rho>0$ such that for every $x \in \Omega$ there is a ball $B_\rho$ of radius $\rho$ such that $x \in B_\rho \subset \Omega$. This rhoughly means that the curvature of the domain is bounded from above.

In some other article of Anzellotti and Giaquinta they prove a similar trace inequality for bounded domains with $C^1$ boundary. My question is:

If a bounded open set $\Omega$ has $C^1$ boundary, is it true that it satisfies the interior sphere condition mentioned above?

[edit] If the answer is negative for $C^1$ boundary, is it possible that for a $C^k$ with $k \geq 2$ or $C^\infty$ boundary the result becomes true?