Gromov product structures, quadrangle structures and split metric decompositions for finite metric spaces

Let (X, d) be a finite metric space with elements P-i, i = 1,..., n and with distances d(ij) := d(P-i, P-j) for i, j = 1,..., n. The "Gromov product" Delta(ijk), is defined as Delta(ijk) = 1/2 (d(ij) + d(ik) - d(jk)). (X, d) is called Delta-generic, if, for each fixed i, the set of Gromov products Delta(ijk) has a unique smallest element, Delta(ijiki). The Gromov product structure on a Delta-generic finite metric space (X, d) is the map that assigns the edge E-jiki to Pi. A finite metric space is called "quadrangle generic", if for all 4-point subsets {P-i, P-j, P-k, P-l}, the set {d(ij)+ d(kl), d(ik)+d(jl), d(il)+ d(jk)} has a unique maximal element. The ``quadrangle structure" on a quadrangle generic finite metric space (X, d) is defined as a map that assigns to each 4-point subset of X the pair of edges corresponding to the maximal element of the sums of distances. Two metric spaces (X, d) and (X, d') are said to be Delta-equivalent (Q-equivalent), if the corresponding Gromov product (quadrangle) structures are the same up to a permutation of X. We show that Gromov product classification is coarser than the metric fan classification. Furthermore it is proved that: (i) The isolation index of the 1-split metric delta(i) is equal to the minimal Gromov product at the vertex Pi. (ii) For a quadrangle generic (X, d), the isolation index of the 2-split metric delta(ij) is nonzero if and only if the edge Eij is a side in every quadrangle whose set of vertices includes Pi and P-j. (iii) For a quadrangle generic (X, d), the isolation index of an m-split metric delta(i1...)i(m) is nonzero if and only if any edge E-ikil is a side in every quadrangle whose vertex set contains P-ik and P-il. These results are applied to construct a totally split decomposable metric for n = 6. (c) 2021 Elsevier B.V. All rights reserved.