{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:LOROMIFVDSKDIJEEXMGBCNWERD","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5e34918560549b28c8e6f7460f2f2b4704b7c69cbcafe32d7add0bd86f8059dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-30T13:08:32Z","title_canon_sha256":"634fb25200df277bd4b13463e20fd1f05bd4c91fe391a7c5d5694c96b106e8e3"},"schema_version":"1.0","source":{"id":"1608.08436","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.08436","created_at":"2026-05-18T01:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"1608.08436v1","created_at":"2026-05-18T01:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08436","created_at":"2026-05-18T01:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"LOROMIFVDSKD","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LOROMIFVDSKDIJEE","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LOROMIFV","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:a8859f236dfe03e7bde293d8f0ba6b49569e44f0ce0dfeb7d88f3cae108da29e","target":"graph","created_at":"2026-05-18T01:07:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this work, we study the volume ratio of the projective tensor products $\\ell^n_p\\otimes_{\\pi}\\ell_q^n\\otimes_{\\pi}\\ell_r^n$ with $1\\leq p\\leq q \\leq r \\leq \\infty$. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kashin type whenever $1\\leq p \\leq q\\leq r \\leq 2$ or $1\\leq p \\leq 2 \\leq r \\leq \\infty$ and $q=2$. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype $2$ constant, we obtain information on the cotype of these $3$-fold proj","authors_text":"Carsten Sch\\\"utt, Elisabeth Werner, Joscha Prochno, Nicole Tomczak-Jaegermann, Ohad Giladi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-30T13:08:32Z","title":"On the geometry of projective tensor products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08436","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bf8ae37d363eb540dd01abf1c029c1fc8a384b2e9983086ebb643b4ba10d9c4a","target":"record","created_at":"2026-05-18T01:07:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5e34918560549b28c8e6f7460f2f2b4704b7c69cbcafe32d7add0bd86f8059dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-30T13:08:32Z","title_canon_sha256":"634fb25200df277bd4b13463e20fd1f05bd4c91fe391a7c5d5694c96b106e8e3"},"schema_version":"1.0","source":{"id":"1608.08436","kind":"arxiv","version":1}},"canonical_sha256":"5ba2e620b51c94342484bb0c1136c488dbdfa64a14f63315761a1ec392990653","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ba2e620b51c94342484bb0c1136c488dbdfa64a14f63315761a1ec392990653","first_computed_at":"2026-05-18T01:07:18.114552Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:18.114552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iJ4aUzuTiTwI6HXY7rGfEio8cbkxT0RFHoIZOIIFzxsB/EG6wFFnd7cWY1lmFLB68WHDzgfHlzk9fqVexh9uDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:18.115127Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.08436","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf8ae37d363eb540dd01abf1c029c1fc8a384b2e9983086ebb643b4ba10d9c4a","sha256:a8859f236dfe03e7bde293d8f0ba6b49569e44f0ce0dfeb7d88f3cae108da29e"],"state_sha256":"e2f9e633af8d0c72896bcd459f1cea4c103a03a5c5b9306bd296e011f8222b28"}