A polygon is a Hamiltonian circuit of the complete graph on n vertices. If we assign real-number ``lengths'' to the edges, each polygon has a length (that is, a real number), which induces a linear quasiordering of the set of all polygons. We call such a quasiordering realizable.
Now suppose the ``lengths'' really are lengths. That is, we pick n points in Euclidean space Ed, (Pi) = (P1, P2, ..., Pn), and define the length of edge ij to be the distance d(Pi, Pj). There are some obvious questions. Which realizable quasiorderings are realizable by points in Ed? One could allow some of the points to coincide, or not; these give different answers. Given points (Pi) , inducing a certain realizable quasiordering, which other realizable quasiorderings are realizable by points (Qi) arbitrarily near (Pi)? I will discuss these questions.
This will be a very informal talk with at most bits of hints of proof.