{"paper":{"title":"The hamburger theorem","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, Mikio Kano","submitted_at":"2015-03-23T21:57:59Z","abstract_excerpt":"We generalize the ham sandwich theorem to $d+1$ measures in $\\mathbb{R}^d$ as follows. Let $\\mu_1,\\mu_2, \\dots, \\mu_{d+1}$ be absolutely continuous finite Borel measures on $\\mathbb{R}^d$. Let $\\omega_i=\\mu_i(\\mathbb{R}^d)$ for $i\\in [d+1]$, $\\omega=\\min\\{\\omega_i; i\\in [d+1]\\}$ and assume that $\\sum_{j=1}^{d+1} \\omega_j=1$. Assume that $\\omega_i \\le 1/d$ for every $i\\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies $\\mu_i(H) \\le (\\sum_{j=1}^{d+1} \\mu_j(H))/d$ for every $i \\in [d+1]$ and $\\sum_{j=1}^{d+1} \\mu_j(H) \\ge \\min(1/2, 1-d\\omega) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06856","kind":"arxiv","version":4},"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"}