{"paper":{"title":"Gromov Product Decomposition of 7-point Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ayse Humeyra Bilge, Metehan Incegul","submitted_at":"2018-04-09T15:11:12Z","abstract_excerpt":"Let $X$ be a finite metric space with elements $P_i$, $i=1,\\dots,n$ and with distance functions $d_{ij}$. The Gromov product of the triangle with vertices $P_i$, $P_j$ and $P_k$ at the vertex $P_i$ is defined by $\\Delta_{ijk}=\\frac{1}{2}(d_{ij}+d_{ik}-d_{jk})$. A metric space is called $\\Delta$-generic, if the set of Gromov products at each $P_i$ has a unique smallest element $\\Delta_{ijk}$.\n  For a $\\Delta$-generic metric space, the map $P_i\\to (P_jP_k)$, where $(P_jP_k)$ is the edge joining $P_j$ to $P_k$ is a well defined map called the \"Gromov product structure\" [Bilge, Celik and Kocak, \"A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03051","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}