{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:VWX7AVUYHAPE2G6EDUXNUC7NZT","short_pith_number":"pith:VWX7AVUY","canonical_record":{"source":{"id":"2411.14129","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-11-21T13:56:33Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"91a6a84820e9369d84bfabe326b250d169ea7e5618e381dafe3fab256e88ff34","abstract_canon_sha256":"72b2c8c65aa5be69bb57ab84f544a56cf7fbc66051190892a350919b311159f3"},"schema_version":"1.0"},"canonical_sha256":"adaff05698381e4d1bc41d2eda0bedccc1430caccfed9d50ca85dc742504d72e","source":{"kind":"arxiv","id":"2411.14129","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.14129","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"arxiv_version","alias_value":"2411.14129v1","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.14129","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"pith_short_12","alias_value":"VWX7AVUYHAPE","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"pith_short_16","alias_value":"VWX7AVUYHAPE2G6E","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"pith_short_8","alias_value":"VWX7AVUY","created_at":"2026-07-05T09:38:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:VWX7AVUYHAPE2G6EDUXNUC7NZT","target":"record","payload":{"canonical_record":{"source":{"id":"2411.14129","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-11-21T13:56:33Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"91a6a84820e9369d84bfabe326b250d169ea7e5618e381dafe3fab256e88ff34","abstract_canon_sha256":"72b2c8c65aa5be69bb57ab84f544a56cf7fbc66051190892a350919b311159f3"},"schema_version":"1.0"},"canonical_sha256":"adaff05698381e4d1bc41d2eda0bedccc1430caccfed9d50ca85dc742504d72e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:38:42.153382Z","signature_b64":"8nhHMM2bjTuNzONT+BwVgZX7Hz15lBW0n3L7CitoTAz8UkyGV7YGxpbFJo8YD280ohCcFdlrSE4weQNIwk9hCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"adaff05698381e4d1bc41d2eda0bedccc1430caccfed9d50ca85dc742504d72e","last_reissued_at":"2026-07-05T09:38:42.152916Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:38:42.152916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.14129","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:38:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WaIQA5kVR8uwaUmnVzWrfPNwoHttRUmIv0Xdo2LGks2KfEOp432PmPlLUyMo+qU7DnOh5aQXzv5jtmme3ZUbDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T02:30:13.119945Z"},"content_sha256":"e655332ec19d19ebe363921c53807b2813f43d77f1be0cde16b7ba7b1deed5d3","schema_version":"1.0","event_id":"sha256:e655332ec19d19ebe363921c53807b2813f43d77f1be0cde16b7ba7b1deed5d3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:VWX7AVUYHAPE2G6EDUXNUC7NZT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On averaged self-distances in finite dimensional Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Gyula Lakos","submitted_at":"2024-11-21T13:56:33Z","abstract_excerpt":"Assume that $\\mathfrak A$ is a real Banach space of finite dimension $n\\geq2$. Consider any Borel probability measure $\\nu$ supported on the unit ball $K$ of $\\mathfrak A$. We show that \\[\\Delta(\\nu)=\\int_{x \\in K}\\int_{ y\\in K}|x-y|_{\\mathfrak A} \\,\\,\\,\\nu(x)\\,\\nu(y)\\leq 2(1-2^{-n}f(n)),\\] where $f:\\mathbb N\\setminus \\{0,1\\}\\rightarrow (0,1]$ is a concrete universal function such that $f(n)\\sim \\frac{2}{\\mathrm e n^2\\log n}$. It is hoped that in the estimate`$f(n)$' can be replaced by `$1$'."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14129","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.14129/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:38:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Usy7D6Kj8cYs+lNgXY61VCGEkn7PeeiKfzj7StzkqrjfmiEY3gDZhVtT3Pm1EPRmz7fZ+Ym65Dlsw1qc4zfPAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T02:30:13.120841Z"},"content_sha256":"3392428dc189ff4f31fe37247b530ec0d6f69fca23ac2bd1379d4f86ec55c0cd","schema_version":"1.0","event_id":"sha256:3392428dc189ff4f31fe37247b530ec0d6f69fca23ac2bd1379d4f86ec55c0cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VWX7AVUYHAPE2G6EDUXNUC7NZT/bundle.json","state_url":"https://pith.science/pith/VWX7AVUYHAPE2G6EDUXNUC7NZT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VWX7AVUYHAPE2G6EDUXNUC7NZT/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-13T02:30:13Z","links":{"resolver":"https://pith.science/pith/VWX7AVUYHAPE2G6EDUXNUC7NZT","bundle":"https://pith.science/pith/VWX7AVUYHAPE2G6EDUXNUC7NZT/bundle.json","state":"https://pith.science/pith/VWX7AVUYHAPE2G6EDUXNUC7NZT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VWX7AVUYHAPE2G6EDUXNUC7NZT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:VWX7AVUYHAPE2G6EDUXNUC7NZT","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":"72b2c8c65aa5be69bb57ab84f544a56cf7fbc66051190892a350919b311159f3","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-11-21T13:56:33Z","title_canon_sha256":"91a6a84820e9369d84bfabe326b250d169ea7e5618e381dafe3fab256e88ff34"},"schema_version":"1.0","source":{"id":"2411.14129","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.14129","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"arxiv_version","alias_value":"2411.14129v1","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.14129","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"pith_short_12","alias_value":"VWX7AVUYHAPE","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"pith_short_16","alias_value":"VWX7AVUYHAPE2G6E","created_at":"2026-07-05T09:38:42Z"},{"alias_kind":"pith_short_8","alias_value":"VWX7AVUY","created_at":"2026-07-05T09:38:42Z"}],"graph_snapshots":[{"event_id":"sha256:3392428dc189ff4f31fe37247b530ec0d6f69fca23ac2bd1379d4f86ec55c0cd","target":"graph","created_at":"2026-07-05T09:38:42Z","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/2411.14129/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Assume that $\\mathfrak A$ is a real Banach space of finite dimension $n\\geq2$. Consider any Borel probability measure $\\nu$ supported on the unit ball $K$ of $\\mathfrak A$. We show that \\[\\Delta(\\nu)=\\int_{x \\in K}\\int_{ y\\in K}|x-y|_{\\mathfrak A} \\,\\,\\,\\nu(x)\\,\\nu(y)\\leq 2(1-2^{-n}f(n)),\\] where $f:\\mathbb N\\setminus \\{0,1\\}\\rightarrow (0,1]$ is a concrete universal function such that $f(n)\\sim \\frac{2}{\\mathrm e n^2\\log n}$. It is hoped that in the estimate`$f(n)$' can be replaced by `$1$'.","authors_text":"Gyula Lakos","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-11-21T13:56:33Z","title":"On averaged self-distances in finite dimensional Banach spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14129","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:e655332ec19d19ebe363921c53807b2813f43d77f1be0cde16b7ba7b1deed5d3","target":"record","created_at":"2026-07-05T09:38:42Z","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":"72b2c8c65aa5be69bb57ab84f544a56cf7fbc66051190892a350919b311159f3","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-11-21T13:56:33Z","title_canon_sha256":"91a6a84820e9369d84bfabe326b250d169ea7e5618e381dafe3fab256e88ff34"},"schema_version":"1.0","source":{"id":"2411.14129","kind":"arxiv","version":1}},"canonical_sha256":"adaff05698381e4d1bc41d2eda0bedccc1430caccfed9d50ca85dc742504d72e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"adaff05698381e4d1bc41d2eda0bedccc1430caccfed9d50ca85dc742504d72e","first_computed_at":"2026-07-05T09:38:42.152916Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:38:42.152916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8nhHMM2bjTuNzONT+BwVgZX7Hz15lBW0n3L7CitoTAz8UkyGV7YGxpbFJo8YD280ohCcFdlrSE4weQNIwk9hCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:38:42.153382Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.14129","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e655332ec19d19ebe363921c53807b2813f43d77f1be0cde16b7ba7b1deed5d3","sha256:3392428dc189ff4f31fe37247b530ec0d6f69fca23ac2bd1379d4f86ec55c0cd"],"state_sha256":"abd8e80e64809649abba234990a714e5da184e7f09de216a6f3bc57795c355de"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wc37AYxxhVmVP6WJJXWguBeqX0y5MHNxFSBaSLsI0pZ0Heawvd1BQtHVL8C/pcjTp9Kn3eYCzZTK9+ta6wWrCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T02:30:13.125860Z","bundle_sha256":"68a665223df027517207409e72a48d0ccc1b390f3b8d95de023dd06e7bc80ea0"}}