Let $c: \mathbb{R} \rightarrow \mathbb{R}^2$ be a simple closed curve with curvature $\kappa \geq 0$.

Then the interior of $c$ is convex.

I know that in this case $$ \langle N(t_0), c(t) - c(t_0) \rangle \geq 0 \quad \text{for all}\; t, t_0 \in \mathbb{R}. $$

But how do I obtain that $\text{Int}(c)$ is convex from there?


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