sufficient and necessary conditions for a set to be a convex hull in $\mathbb{R}^2$

Question

In this question $A\subseteq \mathbb{R}^2$ and by convex hull I mean the smallest (w.r.t. inclusion) convex set $C(A)$ that contains $A$.

Now, let us define: $$A_k=\{\sum_{i=1}^k\lambda_ia_i:\sum_{i=1}^k \lambda_i=1,\lambda_i\in[0,1],a_i\in A\}$$ It is possible to show that $A_3=C(A)$ for any $A$. It is also obvious that $A=A_1=C(A)$ iff $A$ is convex.

My question is, what are the sufficient and\or necessary conditions for $A_2=C(A)$?