Geodesic
Classification of metric spaces whose Steiner--Gromov ratio is equal to one
Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 5, pp. 181-189.

Voir la notice de l'article provenant de la source Math-Net.Ru

Several equivalent conditions for the Steiner–Gromov ratio of a metric space to be equal to one are stated, i.e., conditions for each minimal spanning tree in any finite subset of a given metric space to be both a shortest tree and a minimal filling. A complete classification of such spaces is obtained. 
@article{FPM_2016_21_5_a7,
     author = {A. S. Pahkomova},
     title = {Classification of metric spaces whose {Steiner--Gromov} ratio is equal to one},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {181--189},
     publisher = {mathdoc},
     volume = {21},
     number = {5},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2016_21_5_a7/}
}
TY  - JOUR
AU  - A. S. Pahkomova
TI  - Classification of metric spaces whose Steiner--Gromov ratio is equal to one
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2016
SP  - 181
EP  - 189
VL  - 21
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2016_21_5_a7/
LA  - ru
ID  - FPM_2016_21_5_a7
ER  - 
%0 Journal Article
%A A. S. Pahkomova
%T Classification of metric spaces whose Steiner--Gromov ratio is equal to one
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2016
%P 181-189
%V 21
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2016_21_5_a7/
%G ru
%F FPM_2016_21_5_a7
A. S. Pahkomova. Classification of metric spaces whose Steiner--Gromov ratio is equal to one. Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 5, pp. 181-189. http://geodesic.mathdoc.fr/item/FPM_2016_21_5_a7/

[1] Bednov B. B., Borodin P. A., “Banakhovy prostranstva, realizuyuschie minimalnye zapolneniya”, Matem. sb., 205:4 (2014), 3–20 | DOI | MR | Zbl

[2] Borodin P. A., “Primer nesuschestvovaniya tochki Shteinera v banakhovom prostranstve”, Matem. zametki, 87:4 (2010), 514–518 | DOI | Zbl

[3] Garkavi A. L., Shmatkov V. A., “O tochke Lame i ee obobscheniyakh v normirovannom prostranstve”, Matem. sb., 95 (137):2 (10) (1974), 272–293 | Zbl

[4] Zaretskii K. A., “Postroenie dereva po naboru rasstoyanii mezhdu visyachimi vershinami”, UMN, 20:6 (1965), 90–92 | MR

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

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

[7] Pakhomova A. S., “Otsenki dlya subotnosheniya Shteinera i otnosheniya Shteinera–Gromova”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2014, no. 1, 17–25 | MR | Zbl

[8] Baronti M., Casini E., Papini P. L., “Equilateral sets and their central points”, Rend. Mat. Appl., 13:1 (1993), 133–148 | MR | Zbl

[9] Cieslik D., The Steiner Ratio, Kluwer Academic, Boston, 2001 | MR | Zbl

[10] Gilbert E. N., Pollak H. O., “Steiner minimal trees”, SIAM J. Appl. Math., 16:1 (1968), 1–29 | DOI | MR | Zbl

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

[12] Ivanov A. O., Tuzhilin A. A., “Minimal fillings of finite metric spaces: The state of the art”, Discrete Geometry and Algebraic Combinatorics, Contemp. Math., 625, eds. A. Barg, O. Musin, Amer. Math. Soc., Providence, 2014, 9–35 | DOI | MR | Zbl

[13] Ivanov A. O., Tuzhilin A. A., “The Steiner ratio Gilbert–Pollak Conjecture is still open”, Algorithmica, 62:1–2 (2014), 630–632 | MR

[14] Papini P. L., “Two new examples of sets without medians and centers”, Soc. Estad. Invest. Operat. Top., 13:2 (2005), 315–320 | MR | Zbl

[15] Richmond B., Richmond T., “Metric spaces in which all triangles are degenerate”, Am. Math. Month., 104:8 (1997), 713–719 | DOI | MR | Zbl

[16] Vesely L., “A characterization of reflexivity in the terms of the existence of generalized centers”, Extr. Math., 8:2–3 (1993), 125–131 | MR | Zbl