{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:LTN2RVDWHOYLZNA2VAJBYUHMJ4","short_pith_number":"pith:LTN2RVDW","canonical_record":{"source":{"id":"1007.5101","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-29T02:22:50Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"97afeebb275f3c35e13cfcc5178544edc5ca6f4f80a7f5e0475b8c9fdaed67d6","abstract_canon_sha256":"0452f9ee79316cf468789c8d936c70283854b2bd726edddbce063205af695b7e"},"schema_version":"1.0"},"canonical_sha256":"5cdba8d4763bb0bcb41aa8121c50ec4f3745b92515225f1e5edcae8430972d88","source":{"kind":"arxiv","id":"1007.5101","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.5101","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"arxiv_version","alias_value":"1007.5101v2","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.5101","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"pith_short_12","alias_value":"LTN2RVDWHOYL","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"LTN2RVDWHOYLZNA2","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"LTN2RVDW","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:LTN2RVDWHOYLZNA2VAJBYUHMJ4","target":"record","payload":{"canonical_record":{"source":{"id":"1007.5101","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-29T02:22:50Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"97afeebb275f3c35e13cfcc5178544edc5ca6f4f80a7f5e0475b8c9fdaed67d6","abstract_canon_sha256":"0452f9ee79316cf468789c8d936c70283854b2bd726edddbce063205af695b7e"},"schema_version":"1.0"},"canonical_sha256":"5cdba8d4763bb0bcb41aa8121c50ec4f3745b92515225f1e5edcae8430972d88","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:33.913970Z","signature_b64":"hmZX0QuTJwouy3yrRP9kHDLlpbv5XySFuyxPSTacCN1kKXbJTrqEBD0s6dSn8sM6SIGfY6IMeR6udjx6tqUcCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5cdba8d4763bb0bcb41aa8121c50ec4f3745b92515225f1e5edcae8430972d88","last_reissued_at":"2026-05-18T04:40:33.913519Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:33.913519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.5101","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":"2026-05-18T04:40:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B9Qd4CWV0rBVbbHap4Meo9iJIj/QWj41+dJ4FU2sunClwACw3BNLGcfQgo11VQRoGv4NbF8/OVqDZ0droIBBAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T14:55:51.370594Z"},"content_sha256":"6d57e72442cdffbae2b46686332afbdcf1b5a5fc1d40fac803c43fdf4e304a23","schema_version":"1.0","event_id":"sha256:6d57e72442cdffbae2b46686332afbdcf1b5a5fc1d40fac803c43fdf4e304a23"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:LTN2RVDWHOYLZNA2VAJBYUHMJ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A relative isoperimetric inequality for certain warped product spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Shawn Rafalski","submitted_at":"2010-07-29T02:22:50Z","abstract_excerpt":"Given a warped product space $\\mathbb{R} \\times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a $k$--dimensional region $F \\subset \\{b\\} \\times N$, and the horizontal graph $C \\subset \\mathbb{R} \\times_{f} N$ of an almost everywhere differentiable map over $F$, we prove that the $k$--volume of $C$ is always at least the $k$--volume of the smooth constant height graph ove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.5101","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":"2026-05-18T04:40:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6PwPokV9NseCV4TpMwuX6JgXAZFVvk/YFL12chJWgDzKxizVXF1p6TwPXRZRcCSyK16MaSz/tJDn5dYYOtDGCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T14:55:51.370932Z"},"content_sha256":"6c6e776707c056a07c471ae550819bf161a628a4afee4eae427b10e836fa9774","schema_version":"1.0","event_id":"sha256:6c6e776707c056a07c471ae550819bf161a628a4afee4eae427b10e836fa9774"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LTN2RVDWHOYLZNA2VAJBYUHMJ4/bundle.json","state_url":"https://pith.science/pith/LTN2RVDWHOYLZNA2VAJBYUHMJ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LTN2RVDWHOYLZNA2VAJBYUHMJ4/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":"2026-06-27T14:55:51Z","links":{"resolver":"https://pith.science/pith/LTN2RVDWHOYLZNA2VAJBYUHMJ4","bundle":"https://pith.science/pith/LTN2RVDWHOYLZNA2VAJBYUHMJ4/bundle.json","state":"https://pith.science/pith/LTN2RVDWHOYLZNA2VAJBYUHMJ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LTN2RVDWHOYLZNA2VAJBYUHMJ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:LTN2RVDWHOYLZNA2VAJBYUHMJ4","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":"0452f9ee79316cf468789c8d936c70283854b2bd726edddbce063205af695b7e","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-29T02:22:50Z","title_canon_sha256":"97afeebb275f3c35e13cfcc5178544edc5ca6f4f80a7f5e0475b8c9fdaed67d6"},"schema_version":"1.0","source":{"id":"1007.5101","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.5101","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"arxiv_version","alias_value":"1007.5101v2","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.5101","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"pith_short_12","alias_value":"LTN2RVDWHOYL","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"LTN2RVDWHOYLZNA2","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"LTN2RVDW","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:6c6e776707c056a07c471ae550819bf161a628a4afee4eae427b10e836fa9774","target":"graph","created_at":"2026-05-18T04:40:33Z","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":"Given a warped product space $\\mathbb{R} \\times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a $k$--dimensional region $F \\subset \\{b\\} \\times N$, and the horizontal graph $C \\subset \\mathbb{R} \\times_{f} N$ of an almost everywhere differentiable map over $F$, we prove that the $k$--volume of $C$ is always at least the $k$--volume of the smooth constant height graph ove","authors_text":"Shawn Rafalski","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-29T02:22:50Z","title":"A relative isoperimetric inequality for certain warped product spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.5101","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":"sha256:6d57e72442cdffbae2b46686332afbdcf1b5a5fc1d40fac803c43fdf4e304a23","target":"record","created_at":"2026-05-18T04:40:33Z","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":"0452f9ee79316cf468789c8d936c70283854b2bd726edddbce063205af695b7e","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-07-29T02:22:50Z","title_canon_sha256":"97afeebb275f3c35e13cfcc5178544edc5ca6f4f80a7f5e0475b8c9fdaed67d6"},"schema_version":"1.0","source":{"id":"1007.5101","kind":"arxiv","version":2}},"canonical_sha256":"5cdba8d4763bb0bcb41aa8121c50ec4f3745b92515225f1e5edcae8430972d88","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5cdba8d4763bb0bcb41aa8121c50ec4f3745b92515225f1e5edcae8430972d88","first_computed_at":"2026-05-18T04:40:33.913519Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:33.913519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hmZX0QuTJwouy3yrRP9kHDLlpbv5XySFuyxPSTacCN1kKXbJTrqEBD0s6dSn8sM6SIGfY6IMeR6udjx6tqUcCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:33.913970Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.5101","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d57e72442cdffbae2b46686332afbdcf1b5a5fc1d40fac803c43fdf4e304a23","sha256:6c6e776707c056a07c471ae550819bf161a628a4afee4eae427b10e836fa9774"],"state_sha256":"ee42a4f2718739fc4f0f0a37679e758980c9d3e24a95421f7f4b04efe88e387a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+M6y5M073IC8eirGV4tGD4CBnGATGNCwQJxfAgudriknN4HlcWsE5L9L0FFaVfFcNWUjetVGWEXoso4wpIuABw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T14:55:51.372815Z","bundle_sha256":"dc637e94c5da827e0a5978f6ed057d4abc27dbd24b530b653379eec6871244c0"}}