This question is based on Distributing points evenly on a sphere and Facility location on manifolds
The 'dispersal problem' (which can be mapped to packing disks in many cases) places $n$ points in a region $R$ such that the minimum pair-wise distance among them is maximized.
Question: Does the solution to the dispersal problem with $n$ points in a region $R$ also maximize the length of the shortest Hamiltonian circuit (the optimal traveling salesman trajectory) among any $n$ points in $R$?
Under what conditions on $R$ will this question have a "yes" answer (conditions such as "$R$ has Euclidean metric", "$R$ is a convex planar region" etc.)?
Example: To optimally disperse 5 points on a square, we could place them at the 4 corners and the center — in which case, the best TSP path too appears to be maximally long among all TSP answers with 5 points on the square.
