Five point metric spaces: gromov product structures, quadrangle structures and explicit parameterizations

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” Δ_ijk, is defined as Δ_ijk=1/2(d_ij+d_ik-d_jk). (X,d) is called Δ-generic, if for each fixed i, the set of Gromov products has a unique least element, Δ_(ij_i k_i ). The Gromov product structure on a Δ-generic finite metric space (X,d) is the map that assigns the edge E_(j_i k_i ) to P_i. 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. We define the “quadrangle structure” on a quadrangle generic finite metric space (X,d) as the map that assigns to each 4-point subset of X, the pair of edges corresponding to the maximal element of the sums of the distances. Two metric spaces (X,d) and (X,d') are said to be Δ-equivalent (Q-equivalent), if the corresponding Gromov product (quadrangle) structures are the same, up to a permutation of X. In this paper, Gromov product structures, quadrangle structures, optimal reductions and explicit parameterizations for 5-point spaces are obtained and compared with previous results in the literature. In the final part of this review paper, we have used the Monte Carlo method to obtain the relative volume of each of the 5-point metric types inside the corresponding metric cone for 5-point spaces, meanwhile 102 different partitions of metric cone for 5-point spaces are derived, considering Gromov product structures. These 102 partitions, come in three symmetric classes forming three types of metrics for 5-point spaces. Thus one can say that all the methods of classification given here or given before in the literature of finite metric spaces, give 3 types of metrics for 5-point spaces.