Places where the order of points along a simple polygon passing through them is useful
Your comment that "there are probably too many simple polygons" is the key, because actually there aren't that many.  $n$ points have $n!$ paths (permutations) and $(n-1)!/2$ polygons if we allow self intersections -- that is, $2^{\Theta(n \log n)}$. 

The numbers of paths or polygons without self intersections are $2^{\Theta(n)}$, easiest observed from the fact such paths and polygons can be completed to triangulations, the number of triangulations of a given point set in the plane is $&lt;30^n$ [Sharir-Sheffer'09, latest in a long history], and the number of proper subsets of edges of each triangulation is $&lt;2^{3n-6}$. 

The convex hull of simple polygons has been one of my favorite things since using it to find and/or formulae for polygons in SIGGRAPH'88 http://dx.doi.org/10.1145/54852.378472