Convex curve has convex interior

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?