{"paper":{"title":"Dense point sets with many halving lines","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"G\\'eza T\\'oth, Istv\\'an Kov\\'acs","submitted_at":"2017-04-01T21:00:43Z","abstract_excerpt":"A planar point set of $n$ points is called {\\em $\\gamma$-dense} if the ratio of the largest and smallest distances among the points is at most $\\gamma\\sqrt{n}$. We construct a dense set of $n$ points in the plane with $ne^{\\Omega\\left({\\sqrt{\\log n}}\\right)}$ halving lines. This improves the bound $\\Omega(n\\log n)$ of Edelsbrunner, Valtr and Welzl from 1997.\n  Our construction can be generalized to higher dimensions, for any $d$ we construct a dense point set of $n$ points in $\\mathbb{R}^d$ with $n^{d-1}e^{\\Omega\\left({\\sqrt{\\log n}}\\right)}$ halving hyperplanes. Our lower bounds are asymptoti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.00229","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"}