{"paper":{"title":"Double-normal pairs in the plane and on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"J\\'anos Pach, Konrad J. Swanepoel","submitted_at":"2014-04-09T20:50:41Z","abstract_excerpt":"A double-normal pair of a finite set $S$ of points from Euclidean space is a pair of points $\\{p,q\\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ that are perpendicular to $pq$. A double-normal pair $pq$ is strict if $S\\setminus\\{p,q\\}$ lies in the open strip. We answer a question of Martini and Soltan (2006) by showing that a set of $n\\geq 3$ points in the plane has at most $3\\lfloor n/2\\rfloor$ double-normal pairs. This bound is sharp for each $n\\geq 3$.\n  In a companion paper, we have asymptotically determined this maximum for points in $R^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.2624","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"}