To me it seems possible, to successfully look at symmetric TSP instances from a geometry-point of view.
Examples are:
the diagonals of the convex hull of a set of points in the euclidean plane; those diagonals can't belong to the optimal tour through that points. As this is universally true for all planar euclidean TSP instances, there should be a deeper reason behind it, and, it is also universally true, that
in case of three collinear points, no optimal tour contains the edge connecting the two most distant points of such a triplet.
Another topic is to generalize geometric shapes to symmetric TSP-instances, e.g. how to define subtours with the "shape" of an euclidean circle or, to generalize the shape of planar quadrilaterals to subgraphs induced by 4 vertices (e.g. can the longest diagonal of a convex quadrilateral be part of the optimal tour?)
Questions:
is/was there any research going on in that direction (I'm aware of Linials paper "The geometry of graphs and some of its algorithmic applications", that is based on embedding graphs into euclidean spaces, but not what I am interested in)?
is anyone interested in that topic, which would also cover characterising geometric shapes by length-sums and comparisons alone?
is there already a proper term for coordinate-free, discrete "geometries", that are purely based on length-sums and their comparisons?
Any references or advice would be welcome.
(I was not sure about the proper tagging, so please fix if necessary).