{"paper":{"title":"Topology of the Gr\\\"unbaum-Hadwiger-Ramos hyperplane mass partition problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Albert Haase, Florian Frick, G\\\"unter M. Ziegler, Pavle V. M. Blagojevic","submitted_at":"2015-02-10T16:35:57Z","abstract_excerpt":"In 1960 Gr\\\"unbaum asked whether for any finite mass in $\\mathbb{R}^d$ there are $d$ hyperplanes that cut it into $2^d$ equal parts. This was proved by Hadwiger (1966) for $d\\le3$, but disproved by Avis (1984) for $d\\ge5$, while the case $d=4$ remained open.\n  More generally, Ramos (1996) asked for the smallest dimension $\\Delta(j,k)$ in which for any $j$ masses there are $k$ affine hyperplanes that simultaneously cut each of the masses into $2^k$ equal parts. At present the best lower bounds on $\\Delta(j,k)$ are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.02975","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"}