>>16713160
Hey, I remember getting distracted by this for a couple days a couple years ago. You can get an exact solution for up to k = 5 points and an interval for k > 5, by breaking all the possible shapes down into their component subshapes. For k = 4, use the result of https://doi.org/10.2307/1426176 and permute the order of the points. For k > 4, use https://doi.org/10.1214/16-BJPS315