{"paper":{"title":"Triangulations of monotone families I: Two-dimensional families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.LO"],"primary_cat":"math.AG","authors_text":"Andrei Gabrielov, Nicolai Vorobjov, Saugata Basu","submitted_at":"2014-02-03T18:55:00Z","abstract_excerpt":"Let $K \\subset {\\mathbb R}^n$ be a compact definable set in an o-minimal structure over $\\mathbb R$, e.g., a semi-algebraic or a subanalytic set. A definable family $\\{ S_\\delta|\\> 0< \\delta \\in {\\mathbb R} \\}$ of compact subsets of $K$, is called a monotone family if $S_\\delta \\subset S_\\eta$ for all sufficiently small $\\delta > \\eta >0$. The main result of the paper is that when $\\dim K \\le 2$ there exists a definable triangulation of $K$ such that for each (open) simplex $\\Lambda$ of the triangulation and each small enough $\\delta>0$, the intersection $S_\\delta \\cap \\Lambda$ is equivalent t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0460","kind":"arxiv","version":2},"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"}