Solution:

Let $A_{1} A_{2} \ldots A_{n}$ be a convex $n$-gon. Denote by $\mathcal{B}$ the set of the connected vertices and let $B_{1} B_{2} \ldots B_{k}$ be its convex hull. We shall prove by induction on $n$ that there is a map with the desired properties that in addition sends two given adjacent vertices of the $n$-gon to two given adjacent vertexes of $B_{1} B_{2} \ldots B_{k}$.

The base of the induction $n=3$ is obvious. Suppose that our statement is true for any $k<n$. To prove it for $n$, it is enough to find a map for $n$ that sends $A_{1}$ and $A_{2}$ to $B_{1}$ and $B_{2}$, respectively. Note that there is a unique point $A_{i}$ that is connected with $A_{1}$ and $A_{2}$ (otherwise, some segments will have a common interior point). Consider the points $X_{1}, X_{2}, \ldots, X_{s}$ from $\mathcal{B}$ such that any of the triangles $B_{1} X_{i} B_{2}$ contains no points of $\mathcal{B}$. It is easy to find a point among them, say $X_{l}$, such that the interiors of $\text{B 1 B 2 X l}$ and $\text{B 2 B 1 X l}$ contain at most $n-i$ and $i-3$ points of $\mathcal{B}$, respectively. It is clear now that there are a line through $X_{l}$ and an interior point of the segment $B_{1} B_{2}$ that divides the set $\mathcal{B}$ into two subsets $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, containing $n-i+1$ and $i-2$ points, respectively. Let $B_{1} X_{l}$ and $B_{2} X_{l}$ be sides of the convex hulls of these two sets. If $A_{i}$ is the corresponding point to $X_{l}$, then applying the induction assumption to the sets $A_{2} A_{3} \ldots A_{i}$ and $\mathcal{B}_{2} \cup\{X_{l}\}$, and to $A_{i} A_{i+1} \ldots A_{n} A_{1}$ and $\mathcal{B}_{1} \cup\{X_{l}\}$, we see that the statement is true for $n$ points. This completes the solution of the problem.