Transformations of metrics that preserve the geometric characteristics of finite metric spaces

S. Yu. Lipatov

Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 138-160

Résumé

Given a class $F$ of metric spaces and a family of transformations $T$ of a metric, one has to describe a family of transformations $ T'\subset T$ that transfer $F$ into itself and preserve some types of minimal fillings. The article considers two cases. First, when $F$ is the class of all finite pseudometric spaces, the class $T$ consists of the maps $M\mapsto AM+\tau$, where the matrices $A$ and $\tau$ define the mapping of pseudometric matrix $M$, and the elements of $T'$ preserve any type $G$ of minimal fillings of pseudometric spaces whose points correspond to vertices of degree $1$ of the graph G, and we prove that $A=\lambda E$ for some $\lambda\ge 0$, and $\tau$ is a pseudometric matrix, one of the minimal fillings of which is a star. Second when $F$ is the class of all finite pseudometric spaces, the class $T$ consists of the maps $\rho\to A\rho$, where $A$ is a diagonalizable matrix with two eigenvalues $\lambda_{max}> \lambda_{min} \ge 0$, the largest eigenvalue $\lambda_{max}$ of which has multiplicity $1$, the eigenspace corresponding to the value $ \lambda_{min} $, does not contain nonzero pseudometrics, and the elements of $T'$ preserve the types $G$ of minimal fillings of the pseudometric space, whose points correspond to vertices of degree $1$ of graphs $G$. And we prove that for any mapping matrix from $T'$ there is a pseudometrics that is an eigenvector with the eigenvalue $\lambda_{max}$, among the minimum fillings of which there is a filling of the star type. Second, when $F$ is the class of all finite metric spaces, the class $T$ consists of the maps $\rho\to N\rho$, where the matrix $N$ is the sum of a positive diagonal matrix $A$ and a matrix with the same rows of non-negative elements. The elements of $T'$ preserve all minimal fillings of the type of non-degenerate stars. It has been proven that $T'$ consists of maps $\rho\to N\rho$, where $A$ is scalar. Third, when $F$ is the class of all finite additive metric spaces, $T$ is the class of all linear mappings given by matrices, and the elements of $T'$ preserve all non-degenerate types of minimal fillings, and we proved that for metric spaces consisting of at least four points $T'$ is the set of transformations given by scalar matrices. Fourth, when $F$ is the class of all finite ultrametric spaces, $T$ is the class of all linear mappings given by matrices, and we proved that for three-point spaces the matrices have the form $A=R(B+\lambda E)$, where $B$ is a matrix of identical rows of positive elements, and $R$ is a permutation of the points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$.

Détail

Keywords: minimal fillings, finite pseudometric spaces.