{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:HWH7LBXXC7RDCRGWESPXDBBSGX","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":"1f2e1c2897659b3f3f3685e8aecd06feb99161f0083053b7cfe7a26e8e359fe6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-29T16:05:46Z","title_canon_sha256":"cfe8978045f59e3621004beaf95cd279d71d48593780385d56f2569904495c0e"},"schema_version":"1.0","source":{"id":"1410.8041","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.8041","created_at":"2026-05-18T01:35:40Z"},{"alias_kind":"arxiv_version","alias_value":"1410.8041v2","created_at":"2026-05-18T01:35:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8041","created_at":"2026-05-18T01:35:40Z"},{"alias_kind":"pith_short_12","alias_value":"HWH7LBXXC7RD","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"HWH7LBXXC7RDCRGW","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"HWH7LBXX","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:7ec1a7f03e3f8140a14d9f758fe4a8af1e771937ca54033f76b3626321ca196a","target":"graph","created_at":"2026-05-18T01:35:40Z","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 prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\\in\\re$, $\\Omega\\subset\\re^2$ then the inequality $$\n  \\left(\\frac{|\\Omega|}{\\pi}\\right)^{\\frac{p+1}{2}}\\leq\\frac{1}{2\\pi}\\int_{\\delomega}|x|^pd\\sigma(x) $$ holds true under appropriate assumptions on $\\Omega$ and $p.$ This solves an open problem arising in the context of isoperimetric problems with density and poses some new ones (for instance generalizations to $\\re^n$). We prove the equivalence with a Hardy-Sobolev inequality, giving the best constant, and generalize th","authors_text":"Gyula Csat\\'o","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-29T16:05:46Z","title":"An Isoperimetric Problem With Density and the Hardy Sobolev Inequality in $\\mathbb{R}^2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8041","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:f67ec9abe400163c7c8b1a4c0b777c079f2c2d76f5af9b9e2d9e3868a52c31ad","target":"record","created_at":"2026-05-18T01:35:40Z","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":"1f2e1c2897659b3f3f3685e8aecd06feb99161f0083053b7cfe7a26e8e359fe6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-29T16:05:46Z","title_canon_sha256":"cfe8978045f59e3621004beaf95cd279d71d48593780385d56f2569904495c0e"},"schema_version":"1.0","source":{"id":"1410.8041","kind":"arxiv","version":2}},"canonical_sha256":"3d8ff586f717e23144d6249f71843235f4eb66c2319083a8fcce5a7f43ec7f43","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3d8ff586f717e23144d6249f71843235f4eb66c2319083a8fcce5a7f43ec7f43","first_computed_at":"2026-05-18T01:35:40.070472Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:40.070472Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Eo/gAgMGWwAjSr0PnyGR0/fJbyUq2FNmj6QGE8PqFz/ERe0Lyu7X/5EnMczND6J2wqhvA8xs4XWWXLzeFwoSCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:40.071135Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.8041","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f67ec9abe400163c7c8b1a4c0b777c079f2c2d76f5af9b9e2d9e3868a52c31ad","sha256:7ec1a7f03e3f8140a14d9f758fe4a8af1e771937ca54033f76b3626321ca196a"],"state_sha256":"b940a02eaa0977b51f5aa27f442fb9f799f27f62c5e3cc8d21c7da6aa9714a74"}