Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5776844 | Discrete Mathematics | 2017 | 5 Pages |
Abstract
Let (X,d) be a finite metric space with elements Pi,i=1,â¦,n and with the distance functions dij. The Gromov Product of the “triangle” (Pi,Pj,Pk) with vertices Pi,Pj and Pk at the vertex Pi is defined by Îijk=1â2(dij+dikâdjk). We show that the collection of Gromov products determines the metric. We call a metric space Î-generic, if the set of all Gromov products at a fixed vertex Pi has a unique smallest element (for i=1,â¦,n). We consider the function assigning to each vertex Pi the edge {Pj,Pk} of the triangle (Pi,Pj,Pk) realizing the minimal Gromov product at Pi and we call this function the Gromov product structure of the metric space (X,d). We say two Î-generic metric spaces (X,d) and (X,dâ²) to be Gromov product equivalent, if the corresponding Gromov product structures are the same up to a permutation of X. For n=3,4 there is one (Î-generic) Gromov equivalence class and for n=5 there are three (Î-generic) Gromov equivalence classes. For n=6 we show by computer that there are 26 distinct (Î-generic) Gromov equivalence classes.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
AyÅe Hümeyra Bilge, Derya Ãelik, Åahin Koçak,