Proof of properties of convex polygons

Question

Let $P$ be a polygon on a plane. I want to show the equivalence of the following assertions:

1. $P$ is convex;
2. every internal angle of $P$ is strictly less than $\pi$;
3. $P$ is an intersection of a finite number of closed half-planes;
4. every diagonal of $P$ lies inside of $P$;
5. for every side $AB$ of $P$, $P$ lies in a single half-plane with respect to $AB$.

Some of the implications are trivial, i.e, $1\Longrightarrow 4$, $1\Longrightarrow 2$ and $4 \Longrightarrow 5 $, $3\Longrightarrow 1$.

Surely this question is not new, however I wasn't able to find a reference with a proof.

Answer 1

Found the answer in the book https://bmm.ru/books/details/418737/ (in russian).