{"paper":{"title":"Compact orbit spaces in Hilbert spaces and limits of edge-colouring models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.CO","authors_text":"Alexander Schrijver, Guus Regts","submitted_at":"2012-10-08T09:47:06Z","abstract_excerpt":"Let $G$ be a group of orthogonal transformations of a real Hilbert space $H$. Let $R$ and $W$ be bounded $G$-stable subsets of $H$. Let $\\|.\\|_R$ be the seminorm on $H$ defined by $\\|x\\|_R:=\\sup_{r\\in R}|\\langle r,x\\rangle|$ for $x\\in H$. We show that if $W$ is weakly compact and the orbit space $R^k/G$ is compact for each $k\\in\\oN$, then the orbit space $W/G$ is compact when $W$ is equiped with the norm topology induced by $\\|.\\|_R$.\n  As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lov\\'asz. It forms the edge-colouring counterpart"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2204","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"}