{"paper":{"title":"Topological transversals to a family of convex sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"L. Montejano, R.N. Karasev","submitted_at":"2010-06-01T10:32:21Z","abstract_excerpt":"Let $\\mathcal F$ be a family of compact convex sets in $\\mathbb R^d$. We say that $\\mathcal F $ has a \\emph{topological $\\rho$-transversal of index $(m,k)$} ($\\rho<m$, $0<k\\leq d-m$) if there are, homologically, as many transversal $m$-planes to $\\mathcal F$ as $m$-planes containing a fixed $\\rho$-plane in $\\mathbb R^{m+k}$.\n  Clearly, if $\\mathcal F$ has a $\\rho$-transversal plane, then $\\mathcal F$ has a topological $\\rho$-transversal of index $(m,k),$ for $\\rho<m$ and $k\\leq d-m$. The converse is not true in general.\n  We prove that for a family $\\mathcal F$ of $\\rho+k+1$ compact convex set"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.0104","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"}