{"paper":{"title":"Circles and crossing planar compact convex sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"G\\'abor Cz\\'edli","submitted_at":"2018-02-18T22:26:24Z","abstract_excerpt":"Let $K_0$ be a compact convex subset of the plane $\\mathbb R^2$, and assume that whenever $K_1\\subseteq \\mathbb R^2$ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense due to L. Fejes-T\\'oth. A theorem of L. Fejes-T\\'oth from 1967 states that the assumption above holds for $K_0$ if and only if $K_0$ is a disk. In a paper appeared in 2017, the present author introduced a new concept of crossing, and proved that L. Fejes-T\\'oth's theorem remains true if the old concept is replaced by the new one. Our purpose is to describe the hierarchy among several variants of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.06457","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"}