In the article of Massari presented [here](https://mathoverflow.net/questions/88507/inequality-involving-perimeter-and-area) there is a trace inequality which is said to be true for domains which satisfy the interior sphere condition: 



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



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



Does regularity of the boundary imply interior sphere condition
Elaborating on Malte's answer, it's not the Riemann curvature that matters, it's the second fundamental form of the boundary and, specifically, the reciprocals of its eigenvalues, which are known as the principal radii.


If the boundary is $C^2$, then given any point $x$ on the boundary, there is a positive lower bound $\rho$ for all of the principal radii for points on the boundary within distance $1$ of $x$. Then the ball of radius $\rho$ that is tangent to the boundary at $x$ is contained fully inside the domain.


