{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:UZQA5EEKNYJVM34ZWIEZLP5PML","short_pith_number":"pith:UZQA5EEK","canonical_record":{"source":{"id":"1111.1141","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-04T14:40:54Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"5f5186573dbf0775949f85f99ec918991163fd81db1236c58ea1c4935309366e","abstract_canon_sha256":"f2792c976fcada5f9676b3de4c8e3c844679b31e881ba6ca2944cbccecc9df48"},"schema_version":"1.0"},"canonical_sha256":"a6600e908a6e13566f99b20995bfaf62fd3eb9095f2321b345a2c13eb395f59e","source":{"kind":"arxiv","id":"1111.1141","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.1141","created_at":"2026-05-18T02:39:38Z"},{"alias_kind":"arxiv_version","alias_value":"1111.1141v2","created_at":"2026-05-18T02:39:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.1141","created_at":"2026-05-18T02:39:38Z"},{"alias_kind":"pith_short_12","alias_value":"UZQA5EEKNYJV","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"UZQA5EEKNYJVM34Z","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"UZQA5EEK","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:UZQA5EEKNYJVM34ZWIEZLP5PML","target":"record","payload":{"canonical_record":{"source":{"id":"1111.1141","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-04T14:40:54Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"5f5186573dbf0775949f85f99ec918991163fd81db1236c58ea1c4935309366e","abstract_canon_sha256":"f2792c976fcada5f9676b3de4c8e3c844679b31e881ba6ca2944cbccecc9df48"},"schema_version":"1.0"},"canonical_sha256":"a6600e908a6e13566f99b20995bfaf62fd3eb9095f2321b345a2c13eb395f59e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:38.468855Z","signature_b64":"W7qZmnvq+fN+QQ8l4tU6GhgTE2tFfjMVgk18wpIjiN7RGndC3ChUh5C+xpJ63xjLkgauGuXyooos09HYTzpkBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6600e908a6e13566f99b20995bfaf62fd3eb9095f2321b345a2c13eb395f59e","last_reissued_at":"2026-05-18T02:39:38.468488Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:38.468488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.1141","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:39:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j2RRybbuF+wCb+4VbgVeG+e7B49hAGO4M/zTiHc8d+OoLWt97V8WJZY2g2sJGQe9jjsMuAZzN2xE9CgqXXmVAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:11:34.220285Z"},"content_sha256":"4e9a053d4d6f1ece9d4b294f5b27e9c4ad21af67ea6843341b03c5562bc474a8","schema_version":"1.0","event_id":"sha256:4e9a053d4d6f1ece9d4b294f5b27e9c4ad21af67ea6843341b03c5562bc474a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:UZQA5EEKNYJVM34ZWIEZLP5PML","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal H\\\"older regularity implying finiteness of integral Menger curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Marta Szuma\\'nska, S{\\l}awomir Kolasi\\'nski","submitted_at":"2011-11-04T14:40:54Z","abstract_excerpt":"We study two families of integral functionals indexed by a real number $p > 0$. One family is defined for 1-dimensional curves in $\\R^3$ and the other one is defined for $m$-dimensional manifolds in $\\R^n$. These functionals are described as integrals of appropriate integrands (strongly related to the Menger curvature) raised to power $p$. Given $p > m(m+1)$ we prove that $C^{1,\\alpha}$ regularity of the set (a curve or a manifold), with $\\alpha > \\alpha_0 = 1 - \\frac{m(m+1)}p$ implies finiteness of both curvature functionals ($m=1$ in the case of curves). We also show that $\\alpha_0$ is optim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1141","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:39:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dl+SYnUgJrZnvZioMlcTiSb2bX3xVy8v4qdxrloWB0QANgSGhOET2Nr6S0KULbI9gd3CxlvyBRP1f57rr4W7Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:11:34.220920Z"},"content_sha256":"13963e89e1d238c91920d67f1f69dcbc74ccc94644bdd17aefbcf2dd93ab6da5","schema_version":"1.0","event_id":"sha256:13963e89e1d238c91920d67f1f69dcbc74ccc94644bdd17aefbcf2dd93ab6da5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UZQA5EEKNYJVM34ZWIEZLP5PML/bundle.json","state_url":"https://pith.science/pith/UZQA5EEKNYJVM34ZWIEZLP5PML/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UZQA5EEKNYJVM34ZWIEZLP5PML/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":"2026-06-06T20:11:34Z","links":{"resolver":"https://pith.science/pith/UZQA5EEKNYJVM34ZWIEZLP5PML","bundle":"https://pith.science/pith/UZQA5EEKNYJVM34ZWIEZLP5PML/bundle.json","state":"https://pith.science/pith/UZQA5EEKNYJVM34ZWIEZLP5PML/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UZQA5EEKNYJVM34ZWIEZLP5PML/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:UZQA5EEKNYJVM34ZWIEZLP5PML","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":"f2792c976fcada5f9676b3de4c8e3c844679b31e881ba6ca2944cbccecc9df48","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-04T14:40:54Z","title_canon_sha256":"5f5186573dbf0775949f85f99ec918991163fd81db1236c58ea1c4935309366e"},"schema_version":"1.0","source":{"id":"1111.1141","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.1141","created_at":"2026-05-18T02:39:38Z"},{"alias_kind":"arxiv_version","alias_value":"1111.1141v2","created_at":"2026-05-18T02:39:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.1141","created_at":"2026-05-18T02:39:38Z"},{"alias_kind":"pith_short_12","alias_value":"UZQA5EEKNYJV","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"UZQA5EEKNYJVM34Z","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"UZQA5EEK","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:13963e89e1d238c91920d67f1f69dcbc74ccc94644bdd17aefbcf2dd93ab6da5","target":"graph","created_at":"2026-05-18T02:39:38Z","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":"We study two families of integral functionals indexed by a real number $p > 0$. One family is defined for 1-dimensional curves in $\\R^3$ and the other one is defined for $m$-dimensional manifolds in $\\R^n$. These functionals are described as integrals of appropriate integrands (strongly related to the Menger curvature) raised to power $p$. Given $p > m(m+1)$ we prove that $C^{1,\\alpha}$ regularity of the set (a curve or a manifold), with $\\alpha > \\alpha_0 = 1 - \\frac{m(m+1)}p$ implies finiteness of both curvature functionals ($m=1$ in the case of curves). We also show that $\\alpha_0$ is optim","authors_text":"Marta Szuma\\'nska, S{\\l}awomir Kolasi\\'nski","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-04T14:40:54Z","title":"Minimal H\\\"older regularity implying finiteness of integral Menger curvature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1141","kind":"arxiv","version":2},"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":"sha256:4e9a053d4d6f1ece9d4b294f5b27e9c4ad21af67ea6843341b03c5562bc474a8","target":"record","created_at":"2026-05-18T02:39:38Z","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":"f2792c976fcada5f9676b3de4c8e3c844679b31e881ba6ca2944cbccecc9df48","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-11-04T14:40:54Z","title_canon_sha256":"5f5186573dbf0775949f85f99ec918991163fd81db1236c58ea1c4935309366e"},"schema_version":"1.0","source":{"id":"1111.1141","kind":"arxiv","version":2}},"canonical_sha256":"a6600e908a6e13566f99b20995bfaf62fd3eb9095f2321b345a2c13eb395f59e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a6600e908a6e13566f99b20995bfaf62fd3eb9095f2321b345a2c13eb395f59e","first_computed_at":"2026-05-18T02:39:38.468488Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:39:38.468488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W7qZmnvq+fN+QQ8l4tU6GhgTE2tFfjMVgk18wpIjiN7RGndC3ChUh5C+xpJ63xjLkgauGuXyooos09HYTzpkBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:39:38.468855Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.1141","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4e9a053d4d6f1ece9d4b294f5b27e9c4ad21af67ea6843341b03c5562bc474a8","sha256:13963e89e1d238c91920d67f1f69dcbc74ccc94644bdd17aefbcf2dd93ab6da5"],"state_sha256":"00ffbabf5951d950651524a3f70871cd263fe6ef32f234d064891802b4480e59"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AMBxekQ0czpIpq7MOa6hoOr4R2/6qzroUox1zT2ZQabVM3O4PKdVNVN5sOSP6Cbv9x+iRnq21S5LJF32UtmhBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T20:11:34.223019Z","bundle_sha256":"d2967092e295d63bdfaa974ed6f99767ce505c6a573b5705e012259b0045cb3d"}}