Question 41
Let $n$ be a natural number and $A=\{P_1,\ldots,P_n\}$ a set of $n$ points in the plane, no three collinear. A path through $A$ consists of the $n-1$ segments $P_{\sigma(i)}P_{\sigma(i+1)}$ for $i=1,\ldots,n-1$, where $\sigma$ permutes $\{1,\ldots,n\}$, with no two segments crossing. Prove that the number of different paths through $A$ is minimal if and only if the points form a convex $n$-gon.
