{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:G3UMWXTHQUAVCDFUP3E6PMXLPX","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":"83c275d2ede6afac61a352794989bf2bcf82efa361784da9e600e1a12e5f7a25","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"q-fin.RM","submitted_at":"2024-03-10T12:10:27Z","title_canon_sha256":"a19c93027657e09fc096a276adfd02bfa76bc109645007d4a07eff36b1b70da4"},"schema_version":"1.0","source":{"id":"2403.06188","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.06188","created_at":"2026-07-05T07:54:19Z"},{"alias_kind":"arxiv_version","alias_value":"2403.06188v1","created_at":"2026-07-05T07:54:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.06188","created_at":"2026-07-05T07:54:19Z"},{"alias_kind":"pith_short_12","alias_value":"G3UMWXTHQUAV","created_at":"2026-07-05T07:54:19Z"},{"alias_kind":"pith_short_16","alias_value":"G3UMWXTHQUAVCDFU","created_at":"2026-07-05T07:54:19Z"},{"alias_kind":"pith_short_8","alias_value":"G3UMWXTH","created_at":"2026-07-05T07:54:19Z"}],"graph_snapshots":[{"event_id":"sha256:9ff6903d98ea8e1da656bd036a7195f610ec959003ee2057649328864b8b1e41","target":"graph","created_at":"2026-07-05T07:54:19Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2403.06188/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Geometrically convex functions constitute an interesting class of functions obtained by replacing the arithmetic mean with the geometric mean in the definition of convexity. As recently suggested, geometric convexity may be a sensible property for financial risk measures ([7,13,4]).\n  We introduce a notion of GG-convex conjugate, parallel to the classical notion of convex conjugate introduced by Fenchel, and we discuss its properties. We show how GG-convex conjugation can be axiomatized in the spirit of the notion of general duality transforms introduced in [2,3].\n  We then move to the study o","authors_text":"Fabio Bellini, M\\\"ucahit Ayg\\\"un, Roger J. A. Laeven","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"q-fin.RM","submitted_at":"2024-03-10T12:10:27Z","title":"On Geometrically Convex Risk Measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.06188","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":"sha256:3bb1028cbc86e1dbeea990b4295b0d1851574eb7ae17bb50783358f3c972f012","target":"record","created_at":"2026-07-05T07:54:19Z","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":"83c275d2ede6afac61a352794989bf2bcf82efa361784da9e600e1a12e5f7a25","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"q-fin.RM","submitted_at":"2024-03-10T12:10:27Z","title_canon_sha256":"a19c93027657e09fc096a276adfd02bfa76bc109645007d4a07eff36b1b70da4"},"schema_version":"1.0","source":{"id":"2403.06188","kind":"arxiv","version":1}},"canonical_sha256":"36e8cb5e678501510cb47ec9e7b2eb7ddf4b81fe5d2c7307546d945b721eaf23","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"36e8cb5e678501510cb47ec9e7b2eb7ddf4b81fe5d2c7307546d945b721eaf23","first_computed_at":"2026-07-05T07:54:19.718025Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:54:19.718025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"So/6L4P6ZxytAEsT2HPjaNy2SP7y/NgYtxGMOwEuutFbKYzMM95uv47Se3/XtN4i+Fh7ToJinjE+1HnenyJLDA==","signature_status":"signed_v1","signed_at":"2026-07-05T07:54:19.718411Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.06188","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3bb1028cbc86e1dbeea990b4295b0d1851574eb7ae17bb50783358f3c972f012","sha256:9ff6903d98ea8e1da656bd036a7195f610ec959003ee2057649328864b8b1e41"],"state_sha256":"ccbbefd5965fafd85e828b5de2a20709965ac78b644e9d8d2ccdc790c298c6a8"}