{"paper":{"title":"On three measures of non-convexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Jan Kyn\\v{c}l, Josef Cibulka, Miroslav Korbel\\'a\\v{r}, Pavel Valtr, Rudolf Stola\\v{r}, Viola M\\'esz\\'aros","submitted_at":"2014-10-01T22:57:28Z","abstract_excerpt":"The invisibility graph $I(X)$ of a set $X \\subseteq \\mathbb{R}^d$ is a (possibly infinite) graph whose vertices are the points of $X$ and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in $X$. We consider the following three parameters of a set $X$: the clique number $\\omega(I(X))$, the chromatic number $\\chi(I(X))$ and the convexity number $\\gamma(X)$, which is the minimum number of convex subsets of $X$ that cover $X$. We settle a conjecture of Matou\\v{s}ek and Valtr claiming that for every planar "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.0407","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"}