{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:3BL7CRKWMSZAN5IDOQ4OPMP2YI","short_pith_number":"pith:3BL7CRKW","canonical_record":{"source":{"id":"2506.08803","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-06-10T13:50:11Z","cross_cats_sorted":[],"title_canon_sha256":"d9ac3c76cdef6b6602bea1019d79a6a138bcad397477dc9e18561d43c205b62c","abstract_canon_sha256":"6c6d8e2e628ab8103a8adcaa047c6a25f8935085e420cbd31e39075d2ab3e5dc"},"schema_version":"1.0"},"canonical_sha256":"d857f1455664b206f5037438e7b1fac2361ed76795d6f6717b75b35c36090efd","source":{"kind":"arxiv","id":"2506.08803","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.08803","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"arxiv_version","alias_value":"2506.08803v3","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08803","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"pith_short_12","alias_value":"3BL7CRKWMSZA","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"pith_short_16","alias_value":"3BL7CRKWMSZAN5ID","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"pith_short_8","alias_value":"3BL7CRKW","created_at":"2026-07-05T11:21:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:3BL7CRKWMSZAN5IDOQ4OPMP2YI","target":"record","payload":{"canonical_record":{"source":{"id":"2506.08803","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-06-10T13:50:11Z","cross_cats_sorted":[],"title_canon_sha256":"d9ac3c76cdef6b6602bea1019d79a6a138bcad397477dc9e18561d43c205b62c","abstract_canon_sha256":"6c6d8e2e628ab8103a8adcaa047c6a25f8935085e420cbd31e39075d2ab3e5dc"},"schema_version":"1.0"},"canonical_sha256":"d857f1455664b206f5037438e7b1fac2361ed76795d6f6717b75b35c36090efd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:21:49.030208Z","signature_b64":"xrZu381otPU+X7gqsq3O78N2lGcLIwBKK8SZeXrARoV6ckTL/xlpXFY4dzH1A6NZFEMoszO7f6yHNilz85SsBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d857f1455664b206f5037438e7b1fac2361ed76795d6f6717b75b35c36090efd","last_reissued_at":"2026-07-05T11:21:49.029548Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:21:49.029548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.08803","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:21:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U77kEZyu977M8MvzU0vYdmCDWgDvZrYR1bMiWDa42MGV1CxHiZ3BmyCYwrbayxOOewByAlo5iDCsdsJM2qzlCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T00:41:16.102018Z"},"content_sha256":"f387c474345d30f0d82ae401c97d5007c7f838079a95412061947077ae917bf0","schema_version":"1.0","event_id":"sha256:f387c474345d30f0d82ae401c97d5007c7f838079a95412061947077ae917bf0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:3BL7CRKWMSZAN5IDOQ4OPMP2YI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Anisotropic area measures of convex bodies","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Rolf Schneider","submitted_at":"2025-06-10T13:50:11Z","abstract_excerpt":"Motivated by the relative differential geometry, where the Euclidean normal vector of hypersurfaces is generalized by a relative normalization, we introduce anisotropic area measures of convex bodies, constructed with respect to a gauge body. Together with the anisotropic curvature measures, they are special cases of the newly introduced anisotropic support measures. We show that a convex body in ${\\mathbb R}^n$, for which the anisotropic area measure of some order $k\\in\\{0,\\dots,n-2\\}$ is proportional to the area measure of order $n-1$, must be a $k$-tangential body of the gauge body."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08803","kind":"arxiv","version":3},"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/2506.08803/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":"2026-07-05T11:21:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rYi0Lbrs4OjRyAYmYqfbqkLSmnV6B7SwRmof8wlc2doiWQN/7LZ8wBfG1diTjY0J3RGugmCgx+KGJzGd9b3GBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T00:41:16.102586Z"},"content_sha256":"24b82e98f88b7553f53227df969b688a49f5f145ccf65f62f7dcf7fc4d1d0b16","schema_version":"1.0","event_id":"sha256:24b82e98f88b7553f53227df969b688a49f5f145ccf65f62f7dcf7fc4d1d0b16"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3BL7CRKWMSZAN5IDOQ4OPMP2YI/bundle.json","state_url":"https://pith.science/pith/3BL7CRKWMSZAN5IDOQ4OPMP2YI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3BL7CRKWMSZAN5IDOQ4OPMP2YI/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":"2026-08-09T00:41:16Z","links":{"resolver":"https://pith.science/pith/3BL7CRKWMSZAN5IDOQ4OPMP2YI","bundle":"https://pith.science/pith/3BL7CRKWMSZAN5IDOQ4OPMP2YI/bundle.json","state":"https://pith.science/pith/3BL7CRKWMSZAN5IDOQ4OPMP2YI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3BL7CRKWMSZAN5IDOQ4OPMP2YI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:3BL7CRKWMSZAN5IDOQ4OPMP2YI","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":"6c6d8e2e628ab8103a8adcaa047c6a25f8935085e420cbd31e39075d2ab3e5dc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-06-10T13:50:11Z","title_canon_sha256":"d9ac3c76cdef6b6602bea1019d79a6a138bcad397477dc9e18561d43c205b62c"},"schema_version":"1.0","source":{"id":"2506.08803","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.08803","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"arxiv_version","alias_value":"2506.08803v3","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08803","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"pith_short_12","alias_value":"3BL7CRKWMSZA","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"pith_short_16","alias_value":"3BL7CRKWMSZAN5ID","created_at":"2026-07-05T11:21:49Z"},{"alias_kind":"pith_short_8","alias_value":"3BL7CRKW","created_at":"2026-07-05T11:21:49Z"}],"graph_snapshots":[{"event_id":"sha256:24b82e98f88b7553f53227df969b688a49f5f145ccf65f62f7dcf7fc4d1d0b16","target":"graph","created_at":"2026-07-05T11:21:49Z","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/2506.08803/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Motivated by the relative differential geometry, where the Euclidean normal vector of hypersurfaces is generalized by a relative normalization, we introduce anisotropic area measures of convex bodies, constructed with respect to a gauge body. Together with the anisotropic curvature measures, they are special cases of the newly introduced anisotropic support measures. We show that a convex body in ${\\mathbb R}^n$, for which the anisotropic area measure of some order $k\\in\\{0,\\dots,n-2\\}$ is proportional to the area measure of order $n-1$, must be a $k$-tangential body of the gauge body.","authors_text":"Rolf Schneider","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-06-10T13:50:11Z","title":"Anisotropic area measures of convex bodies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08803","kind":"arxiv","version":3},"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:f387c474345d30f0d82ae401c97d5007c7f838079a95412061947077ae917bf0","target":"record","created_at":"2026-07-05T11:21:49Z","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":"6c6d8e2e628ab8103a8adcaa047c6a25f8935085e420cbd31e39075d2ab3e5dc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-06-10T13:50:11Z","title_canon_sha256":"d9ac3c76cdef6b6602bea1019d79a6a138bcad397477dc9e18561d43c205b62c"},"schema_version":"1.0","source":{"id":"2506.08803","kind":"arxiv","version":3}},"canonical_sha256":"d857f1455664b206f5037438e7b1fac2361ed76795d6f6717b75b35c36090efd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d857f1455664b206f5037438e7b1fac2361ed76795d6f6717b75b35c36090efd","first_computed_at":"2026-07-05T11:21:49.029548Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:21:49.029548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xrZu381otPU+X7gqsq3O78N2lGcLIwBKK8SZeXrARoV6ckTL/xlpXFY4dzH1A6NZFEMoszO7f6yHNilz85SsBA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:21:49.030208Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.08803","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f387c474345d30f0d82ae401c97d5007c7f838079a95412061947077ae917bf0","sha256:24b82e98f88b7553f53227df969b688a49f5f145ccf65f62f7dcf7fc4d1d0b16"],"state_sha256":"b63dcb06b389cccd1ae7e9874ec6f1447796baffc00cfd8967211fdebb19427a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FATaq4oHwN1kJnwoFpfcWM8lkit57mMwhrHDi63guWoUSfH/ciBNEQyEGaDVsTVNw9kGP3hPlxcD8zOCTx1fBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T00:41:16.107110Z","bundle_sha256":"28ba032c36f5e8fb1d726a1391edd9b1274aaeed4830f79f0be317c6ddf9e81f"}}