{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:E5HOLONTN5G3O2LNIL5FE4QCID","short_pith_number":"pith:E5HOLONT","canonical_record":{"source":{"id":"1606.02195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-07T16:09:10Z","cross_cats_sorted":[],"title_canon_sha256":"95f29baf5d444169a1e11bd35a398b69ddd8ec7f1ff240d7ed91f3372745befc","abstract_canon_sha256":"1693592202f79ab42d89ed9212093e721ae9a21c365626e287886b78e70efdb1"},"schema_version":"1.0"},"canonical_sha256":"274ee5b9b36f4db7696d42fa52720240e687b55169c63592858aeafcc420806e","source":{"kind":"arxiv","id":"1606.02195","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02195","created_at":"2026-05-18T01:12:01Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02195v2","created_at":"2026-05-18T01:12:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02195","created_at":"2026-05-18T01:12:01Z"},{"alias_kind":"pith_short_12","alias_value":"E5HOLONTN5G3","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"E5HOLONTN5G3O2LN","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"E5HOLONT","created_at":"2026-05-18T12:30:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:E5HOLONTN5G3O2LNIL5FE4QCID","target":"record","payload":{"canonical_record":{"source":{"id":"1606.02195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-07T16:09:10Z","cross_cats_sorted":[],"title_canon_sha256":"95f29baf5d444169a1e11bd35a398b69ddd8ec7f1ff240d7ed91f3372745befc","abstract_canon_sha256":"1693592202f79ab42d89ed9212093e721ae9a21c365626e287886b78e70efdb1"},"schema_version":"1.0"},"canonical_sha256":"274ee5b9b36f4db7696d42fa52720240e687b55169c63592858aeafcc420806e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:01.286569Z","signature_b64":"/tlnTtY2CQkoJ0pMIHCR9AgntTSzzw6wr7ECTEndDlfe/DiIc4+07FcjfelMZ1Tbcn2jtUgitWUtyxyJp/WtAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"274ee5b9b36f4db7696d42fa52720240e687b55169c63592858aeafcc420806e","last_reissued_at":"2026-05-18T01:12:01.286218Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:01.286218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.02195","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":"2026-05-18T01:12:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ADGerSylKyIa77J9wZogSa5dmQl4NpjnefBT3vQPPK8AmmlOnOBWawaK8gRITTrvCuLjStUm5cmbfgJVzYHZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T03:15:26.655365Z"},"content_sha256":"42302dd0265b7abc82fa5121bf3b9d7dfab539dc96fa75f05088b1b5def70149","schema_version":"1.0","event_id":"sha256:42302dd0265b7abc82fa5121bf3b9d7dfab539dc96fa75f05088b1b5def70149"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:E5HOLONTN5G3O2LNIL5FE4QCID","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some isoperimetric inequalities on $\\mathbb{R} ^N$ with respect to weights $|x|^\\alpha $","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"A. Alvino, A. Mercaldo, F. Brock, F. Chiacchio, M.R. Posteraro","submitted_at":"2016-06-07T16:09:10Z","abstract_excerpt":"We solve a class of isoperimetric problems on $\\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\\in [0,1]$, then among all smooth sets $\\Omega$ in $\\mathbb{R} ^N$ with fixed Lebesgue measure, $\\int_{\\partial \\Omega } |x|^k \\, \\mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Sz\\\"ego principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02195","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":"2026-05-18T01:12:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4FxYIbJrF+RoYpGbc+zc+pewyso9OtRBkiM3oNttdyGnmfXifqs1GY1XX2BPnGjCZPzMIyW2krO+6zzA/HfpAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T03:15:26.655729Z"},"content_sha256":"3ab18e0fce19806a57f5e5ccaab29e8b957ed8bc8c4acce35abed87fcd8bb504","schema_version":"1.0","event_id":"sha256:3ab18e0fce19806a57f5e5ccaab29e8b957ed8bc8c4acce35abed87fcd8bb504"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E5HOLONTN5G3O2LNIL5FE4QCID/bundle.json","state_url":"https://pith.science/pith/E5HOLONTN5G3O2LNIL5FE4QCID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E5HOLONTN5G3O2LNIL5FE4QCID/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":"2026-06-22T03:15:26Z","links":{"resolver":"https://pith.science/pith/E5HOLONTN5G3O2LNIL5FE4QCID","bundle":"https://pith.science/pith/E5HOLONTN5G3O2LNIL5FE4QCID/bundle.json","state":"https://pith.science/pith/E5HOLONTN5G3O2LNIL5FE4QCID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E5HOLONTN5G3O2LNIL5FE4QCID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:E5HOLONTN5G3O2LNIL5FE4QCID","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":"1693592202f79ab42d89ed9212093e721ae9a21c365626e287886b78e70efdb1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-07T16:09:10Z","title_canon_sha256":"95f29baf5d444169a1e11bd35a398b69ddd8ec7f1ff240d7ed91f3372745befc"},"schema_version":"1.0","source":{"id":"1606.02195","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02195","created_at":"2026-05-18T01:12:01Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02195v2","created_at":"2026-05-18T01:12:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02195","created_at":"2026-05-18T01:12:01Z"},{"alias_kind":"pith_short_12","alias_value":"E5HOLONTN5G3","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"E5HOLONTN5G3O2LN","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"E5HOLONT","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:3ab18e0fce19806a57f5e5ccaab29e8b957ed8bc8c4acce35abed87fcd8bb504","target":"graph","created_at":"2026-05-18T01:12:01Z","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 solve a class of isoperimetric problems on $\\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\\in [0,1]$, then among all smooth sets $\\Omega$ in $\\mathbb{R} ^N$ with fixed Lebesgue measure, $\\int_{\\partial \\Omega } |x|^k \\, \\mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Sz\\\"ego principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.","authors_text":"A. Alvino, A. Mercaldo, F. Brock, F. Chiacchio, M.R. Posteraro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-07T16:09:10Z","title":"Some isoperimetric inequalities on $\\mathbb{R} ^N$ with respect to weights $|x|^\\alpha $"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02195","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":"sha256:42302dd0265b7abc82fa5121bf3b9d7dfab539dc96fa75f05088b1b5def70149","target":"record","created_at":"2026-05-18T01:12:01Z","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":"1693592202f79ab42d89ed9212093e721ae9a21c365626e287886b78e70efdb1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-07T16:09:10Z","title_canon_sha256":"95f29baf5d444169a1e11bd35a398b69ddd8ec7f1ff240d7ed91f3372745befc"},"schema_version":"1.0","source":{"id":"1606.02195","kind":"arxiv","version":2}},"canonical_sha256":"274ee5b9b36f4db7696d42fa52720240e687b55169c63592858aeafcc420806e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"274ee5b9b36f4db7696d42fa52720240e687b55169c63592858aeafcc420806e","first_computed_at":"2026-05-18T01:12:01.286218Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:01.286218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/tlnTtY2CQkoJ0pMIHCR9AgntTSzzw6wr7ECTEndDlfe/DiIc4+07FcjfelMZ1Tbcn2jtUgitWUtyxyJp/WtAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:01.286569Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.02195","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42302dd0265b7abc82fa5121bf3b9d7dfab539dc96fa75f05088b1b5def70149","sha256:3ab18e0fce19806a57f5e5ccaab29e8b957ed8bc8c4acce35abed87fcd8bb504"],"state_sha256":"aa096ab909b52373a2fe1d5024abd4c2cf9076f7287f503674140e5fee510fe8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TYnj3SqFzjWiA8aY7XkrpyDd1kTf9WIz8bst/DIRIs6YvpHtNO1N1vJd/RBqZRvxh+bC+mHw78AxogvUzpMDAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T03:15:26.657623Z","bundle_sha256":"c6c1099c4d10435aa07860cb72a4096f6b8f0ae7ca6779b9655d33894d9899ca"}}