{"paper":{"title":"Sufficiency of simplex inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Shuzo Izumi","submitted_at":"2013-09-18T04:16:14Z","abstract_excerpt":"Let z_0,...,z_n be the (n-1)-dimensional volumes of facets of an n-simplex. Then we have the simplex inequalities: z_p < z_0+...+\\check{z}_p+...+z_n (0 =< p =< n), generalizations of triangle inequalities. Conversely, suppose that numbers z_0,...,z_n > 0 satisfy these inequalities. Does there exist an n-simplex the volumes of whose facets are them? Kakeya solved this problem affirmatively in the case n = 3 and conjectured that the assertion is affirmative also for all n >= 4. We prove that his conjecture is affirmative. To do this, we define three kinds of spaces of loops associated to n-simpl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4534","kind":"arxiv","version":3},"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"}