An equivalence class decomposition of finite metric spaces via Gromov products

Let (X,d) be a finite metric space with elements Pi,i=1,…,n and with the distance functions dij. The Gromov Product of the “triangle” (Pi,Pj,Pk) with vertices Pi,Pj and Pk at the vertex Pi is defined by Δijk=1∕2(dij+dik−djk). We show that the collection of Gromov products determines the metric. We call a metric space Δ-generic, if the set of all Gromov products at a fixed vertex Pi has a unique smallest element (for i=1,…,n). We consider the function assigning to each vertex Pi the edge {Pj,Pk} of the triangle (Pi,Pj,Pk) realizing the minimal Gromov product at Pi and we call this function the Gromov product structure of the metric space (X,d). We say two Δ-generic metric spaces (X,d) and (X,d′) to be Gromov product equivalent, if the corresponding Gromov product structures are the same up to a permutation of X. For n=3,4 there is one (Δ-generic) Gromov equivalence class and for n=5 there are three (Δ-generic) Gromov equivalence classes. For n=6 we show by computer that there are 26 distinct (Δ-generic) Gromov equivalence classes.