Combinatorial invariants of finite metric spaces and the Wasserstein arrangement

In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport and are called “Wasserstein polytopes” or “Kantorovich-Rubinstein polytopes” in the literature. Recently such polytopes have been shown to play an important role in a host of different contexts – however, little is known to date about their structure.

In particular, Vershik asked about the stratification of the metric cone according to the combinatorial type of such polytopes.

After stating the definitions and some examples, in this talk I will define an arrangement of hyperplanes that describes the stratification sought by Vershik, together with some computational results on enumerative invariants in the case of metrics on up to six points.

This will also allow us to compare Wasserstein polytopes with "Tight spans”, showing that the stratifications of the metric cone induced by these two combinatorial invariants are not related by refinement. Time permitting, I will mention some open problems.

The talk is based on joint work with Lukas Kühne and Leonie Mühlherr.