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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1309.4534","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-09-18T04:16:14Z","cross_cats_sorted":[],"title_canon_sha256":"29ee4032719b1329f713aa84b70f25107fbcfd4d0757269275f3bbedaaab7001","abstract_canon_sha256":"6e96b5490187f979d87face8de3c15bb4a1a79213f80e2c8a77e3f3010a6dbb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:29.527274Z","signature_b64":"Rgyg7QT/J8bHNhSoJ2pZB3gUEhPB/yN5VSsKg9FWfX9LamPMiPMf3eBVm8ebFKe1D1lNPuf/bYQYcpUXD47gCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3875dfd884bfef229a0e885db703d0e0ca5ddc0a0b6993d2c2d67446668f6ac5","last_reissued_at":"2026-05-18T02:27:29.526403Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:29.526403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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