The definitions I use are the following:
Definition by supporting lines (Convex Curve): A convex curve is a curve in the Euclidean plane ($\mathbb{R}^2$) which lies completely on one side of each and every one of its tangent lines (Wikipedia).
and
Definition (Convex Set): A convex set $A \subseteq \mathbb{R}^n$ is a set of points such that, given any two points $v, w \in A$, the line-segment $[v,w]$ lies entirely within that set (Wikibooks).
I want to prove that given a bounded set $D \subseteq \mathbb{R}^2$ with a smooth boundary, that $$ \text{$D$ is convex}\iff \text{$\partial D$ is a convex curve} $$
where $\partial D = \overline{D} \backslash D^{\circ}$ is the boundary of $D$.
I am aware of the fact that there are two other posts highlighting (link 1, link 2) the same problem (maybe with some differences in definitions), but the way the proofs/questions were formulated were pretty vague.
For the longest time I haven't been able to figure out a formal proof, while it is tauntingly intuitive. Do you guys have any suggestions? Thanks in advance.
P.S. I might need some extra conditions that I have overlooked.