{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QIKNMYFVTP72MGABAEJGISTJ6J","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":"6c7242808e80c29961b9f51f155bda1f32d588d40f16e563b17169535d009ecf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-16T13:32:33Z","title_canon_sha256":"fb739ece7370f22bc8f63bc92739425bd406230c60419b1057023ef089be62e4"},"schema_version":"1.0","source":{"id":"1507.04563","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04563","created_at":"2026-05-18T01:29:53Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04563v3","created_at":"2026-05-18T01:29:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04563","created_at":"2026-05-18T01:29:53Z"},{"alias_kind":"pith_short_12","alias_value":"QIKNMYFVTP72","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QIKNMYFVTP72MGAB","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QIKNMYFV","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:677ca6f92cc0e97a0a6207e716d13757498c99263e42a77a172378f3f2758b29","target":"graph","created_at":"2026-05-18T01:29:53Z","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 present the proof of several inequalities using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, we give a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric ","authors_text":"Xavier Cabre","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-16T13:32:33Z","title":"Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: a survey"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04563","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":"sha256:5d3e468faac40c91408b996e47240009eceacaf0a4a49659c73d6cee5d2b0ed2","target":"record","created_at":"2026-05-18T01:29:53Z","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":"6c7242808e80c29961b9f51f155bda1f32d588d40f16e563b17169535d009ecf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-16T13:32:33Z","title_canon_sha256":"fb739ece7370f22bc8f63bc92739425bd406230c60419b1057023ef089be62e4"},"schema_version":"1.0","source":{"id":"1507.04563","kind":"arxiv","version":3}},"canonical_sha256":"8214d660b59bffa618010112644a69f24540407ce076b078efe16d355a3a82c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8214d660b59bffa618010112644a69f24540407ce076b078efe16d355a3a82c8","first_computed_at":"2026-05-18T01:29:53.875792Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:29:53.875792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TGQ2KO6NBBzT8fM3K4j/HF0TmXnL7mQoi8f/v3SXPUMoIDBrp47Np3hU6jSM1iYDJKRAE2R+LpCRv5LfPTOIAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:29:53.876304Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.04563","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d3e468faac40c91408b996e47240009eceacaf0a4a49659c73d6cee5d2b0ed2","sha256:677ca6f92cc0e97a0a6207e716d13757498c99263e42a77a172378f3f2758b29"],"state_sha256":"047ad428e41bcfa39dcbec1cdb34b7d86893e278706d5cf53ae8f4cdf6275120"}