Gromov Product Decomposition of 7-point Metric Spaces
Ayse Humeyra Bilge, Metehan Incegul

TL;DR
This paper systematically classifies 7-point finite metric spaces using Gromov product decomposition, extending previous classifications for smaller point sets and identifying 431 equivalence classes.
Contribution
It provides the first comprehensive Gromov product-based classification of 7-point metric spaces, expanding the understanding of metric space structures.
Findings
Identified 431 $ riangle$-equivalence classes for 7-point spaces
Extended the classification framework from 5 and 6 points to 7 points
Demonstrated the systematic decomposition method for finite metric spaces
Abstract
Let be a finite metric space with elements , and with distance functions . The Gromov product of the triangle with vertices , and at the vertex is defined by . A metric space is called -generic, if the set of Gromov products at each has a unique smallest element . For a -generic metric space, the map , where is the edge joining to is a well defined map called the "Gromov product structure" [Bilge, Celik and Kocak, "An equivalence class decomposition of finite metric spaces", Discrete Mathmetics, Vol 340, (2017) 1928-1932]. For n=5, the 3 -equivalence classes coincide with the classification of -point metrics. For , there are 26 -equivalence classes refined by 339 metric classes. In this…
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Taxonomy
TopicsDigital Image Processing Techniques · Fuzzy and Soft Set Theory
