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.

Bibliographie

[1] Banks W. D., Conflitti A., Shparlinski I. E., “Character sums over integers with restricted $g$-ary digits”, Illinois J. Math., 46:3 (2002), 819–836 | DOI | MR | Zbl

[2] Ivanov A.O., Tuzhilin A.A., “Odnomernaya problema Gromova o minimalnom zapolnenii”, Matem. sb., 203:5 (2012), 65–118 | DOI | MR | Zbl

[3] Ivanov A.O., Tuzhilin A.A., Teoriya ekstremalnykh setei, IKI, Izhevsk, 424 pp.

[4] Ivanov A.O., Tuzhilin A.A., Minimal Networks. Steiner Problem and Its Generalizations, CRC Press, 1994 | MR | Zbl

[5] Lipatov S.Yu., “Funktsii, ne menyayuschie tipy minimalnykh zapolnenii”, Vestnik Moskovskogo universiteta. Seriya 1: Matematika. Mekhanika, 2015, 42–45 | Zbl

[6] S. Yu Lipatov, “Metrics transformations preserving the types of one-dimensional minimal fillings”, Filomat, 33:4 (2019), 1081–1089 | DOI | MR | Zbl

[7] Moscow Univ. Math. Bull., 67:2 (2012), 52–54 | DOI | MR | Zbl

[8] Z.N.Ovsyannikov, “Otkrytoe semeistvo mnozhestv, dlya kotorykh minimalnoe zapolnenie ne edinstvenno”, Fund. i prikl. matem., 18:2 (2013), 153–156

[9] I. L. Laut, Z. N. Ovsyannikov, “Vid minimalnykh razvetvlennykh geodezicheskikh v normirovannom prostranstve opredelyaet normu”, Fund. i prikl. matem., 18:2 (2013), 67–77

[10] V. A. Mischenko, “Otsenki otnosheniya Shteinera–Gromova rimanovykh mnogoobrazii”, Fund. i prikl. matem., 18:2 (2013), 119–124

[11] Z. N. Ovsyannikov, “Otnosheniya Shteinera, Shteinera–Gromova i subotnosheniya Shteinera dlya prostranstva kompaktov v evklidovoi ploskosti s rasstoyaniem Khausdorfa”, Fund. i prikl. matem., 18:2 (2013), 157–165

[12] Z. N. Ovsyannikov, “Subotnoshenie Shteinera dlya pyati tochek na ploskosti i chetyrekh tochek v prostranstve”, Fund. i prikl. matem., 18:2 (2013), 167–179

[13] A.S.Pakhomova, “Kriterii nepreryvnosti otnoshenii tipa Shteinera v prostranstve Gromova-Khausdorfa”, Matem. zametki, 96:1 (2014), 126–137 | DOI | MR | Zbl

[14] Ivanov A. O., Tuzhilin A. A., “Geometriya minimalnykh setei i odnomernaya problema Plato”, Uspekhi matematicheskikh nauk, 47:2 (1992)

[15] A.Ivanov, and A.Tuzhilin, “Minimal fillings of finite metric spaces: The state of the art”, Contemporary Mathematics, 625, 2014, 9–35 | DOI | MR | Zbl

[16] A. Ivanov, Z. Ovsyannikov, N. Strelkova, A. Tuzhilin, “Odnomernye minimalnye zapolneniya s rebrami otritsatelnogo vesa”, Vestnik Moskovskogo universiteta. Seriya 1. Matematika i mekhanika, 2012, no. 5, 3–8

[17] A. Yu. Eremin, “Formula vesa minimalnogo zapolneniya konechnogo metricheskogo prostranstva”, Matem. sb., 204:9 (2013), 51–72 | DOI | MR | Zbl

[18] D. Z. Du, F. K. Hwang, “A proof of Gilbert-Pollak Conjecture on the Steiner ratio”, Algorithmica, 7 (1992), 121–135 ; Матем. сб., 204:9 (2013), 51–72 | DOI | MR | Zbl | DOI | MR

[19] M. Gromov, “Filling Riemannian manifolds”, J. Diff. Geom., 18:1 (1983), 1–147 | MR | Zbl

[20] D.Yu. Burago, Yu. D. Burago, S. V. Ivanov, Kurs metricheskoi geometrii, In-t kompyuternykh issledovanii, M.-Izhevsk, 2004, 512 pp. | MR