{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:4LJ2FZRB4WEEHREZ6CMEI3LSZB","short_pith_number":"pith:4LJ2FZRB","canonical_record":{"source":{"id":"1009.2218","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2010-09-12T05:56:03Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"a79e2787f61bdbd512b7ca647239d687aadd161cdebc7e82582aa3adebe940e0","abstract_canon_sha256":"d668fbb8741b210d5886a4b8a7ff13652501429ccf8df33d06ece740578de63a"},"schema_version":"1.0"},"canonical_sha256":"e2d3a2e621e58843c499f098446d72c87d8fc9db416e232f9845bfd0bd014f7e","source":{"kind":"arxiv","id":"1009.2218","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2218","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2218v2","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2218","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"pith_short_12","alias_value":"4LJ2FZRB4WEE","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"4LJ2FZRB4WEEHREZ","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"4LJ2FZRB","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:4LJ2FZRB4WEEHREZ6CMEI3LSZB","target":"record","payload":{"canonical_record":{"source":{"id":"1009.2218","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2010-09-12T05:56:03Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"a79e2787f61bdbd512b7ca647239d687aadd161cdebc7e82582aa3adebe940e0","abstract_canon_sha256":"d668fbb8741b210d5886a4b8a7ff13652501429ccf8df33d06ece740578de63a"},"schema_version":"1.0"},"canonical_sha256":"e2d3a2e621e58843c499f098446d72c87d8fc9db416e232f9845bfd0bd014f7e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:55.728565Z","signature_b64":"mpKjjobUn+FZBlK5vVQhpWuE4e1rKd7h2Hcirp+w9g7KG7OeYBOTEe3UualizGUK2R9qwJWDfMMVVSnkzOLTBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2d3a2e621e58843c499f098446d72c87d8fc9db416e232f9845bfd0bd014f7e","last_reissued_at":"2026-05-18T04:40:55.728014Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:55.728014Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.2218","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":"2026-05-18T04:40:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oRazPq4iHj/Vl/WW/o+VXEWpDWGf1fiumHok/kXysYiDPFvPuOr8d0FG28qQKCPCW8ERJPpl1gFK5pjclCZjCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T01:12:39.808691Z"},"content_sha256":"f014b34fe5abe014d4a1b0598412f9faf7d9dc1adf06ece1c90814e94f36d5c3","schema_version":"1.0","event_id":"sha256:f014b34fe5abe014d4a1b0598412f9faf7d9dc1adf06ece1c90814e94f36d5c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:4LJ2FZRB4WEEHREZ6CMEI3LSZB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Isosceles Triangles and Related Problems in a Convex Polygon","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.CG","authors_text":"Amol Aggarwal","submitted_at":"2010-09-12T05:56:03Z","abstract_excerpt":"Given any convex $n$-gon, in this article, we: (i) prove that its vertices can form at most $n^2/2 + \\Theta(n\\log n)$ isosceles trianges with two sides of unit length and show that this bound is optimal in the first order, (ii) conjecture that its vertices can form at most $3n^2/4 + o(n^2)$ isosceles triangles and prove this conjecture for a special group of convex $n$-gons, (iii) prove that its vertices can form at most $\\lfloor n/k \\rfloor$ regular $k$-gons for any integer $k\\ge 4$ and that this bound is optimal, and (iv) provide a short proof that the sum of all the distances between its ve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2218","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":"2026-05-18T04:40:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JiP/Pb7zn+Pok60kmH7NcmbhuOJO4GHT9tSgjxGI11qZdKYJPdSIkANv6piTlP57nxM6wsgLlwMywGuujKFGDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T01:12:39.809040Z"},"content_sha256":"9150d22b7471afc8475854811fe2f3faa9c0735030ae89284935a45b6493c9a1","schema_version":"1.0","event_id":"sha256:9150d22b7471afc8475854811fe2f3faa9c0735030ae89284935a45b6493c9a1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4LJ2FZRB4WEEHREZ6CMEI3LSZB/bundle.json","state_url":"https://pith.science/pith/4LJ2FZRB4WEEHREZ6CMEI3LSZB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4LJ2FZRB4WEEHREZ6CMEI3LSZB/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":"2026-06-03T01:12:39Z","links":{"resolver":"https://pith.science/pith/4LJ2FZRB4WEEHREZ6CMEI3LSZB","bundle":"https://pith.science/pith/4LJ2FZRB4WEEHREZ6CMEI3LSZB/bundle.json","state":"https://pith.science/pith/4LJ2FZRB4WEEHREZ6CMEI3LSZB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4LJ2FZRB4WEEHREZ6CMEI3LSZB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:4LJ2FZRB4WEEHREZ6CMEI3LSZB","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":"d668fbb8741b210d5886a4b8a7ff13652501429ccf8df33d06ece740578de63a","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2010-09-12T05:56:03Z","title_canon_sha256":"a79e2787f61bdbd512b7ca647239d687aadd161cdebc7e82582aa3adebe940e0"},"schema_version":"1.0","source":{"id":"1009.2218","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2218","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2218v2","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2218","created_at":"2026-05-18T04:40:55Z"},{"alias_kind":"pith_short_12","alias_value":"4LJ2FZRB4WEE","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"4LJ2FZRB4WEEHREZ","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"4LJ2FZRB","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:9150d22b7471afc8475854811fe2f3faa9c0735030ae89284935a45b6493c9a1","target":"graph","created_at":"2026-05-18T04:40:55Z","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 any convex $n$-gon, in this article, we: (i) prove that its vertices can form at most $n^2/2 + \\Theta(n\\log n)$ isosceles trianges with two sides of unit length and show that this bound is optimal in the first order, (ii) conjecture that its vertices can form at most $3n^2/4 + o(n^2)$ isosceles triangles and prove this conjecture for a special group of convex $n$-gons, (iii) prove that its vertices can form at most $\\lfloor n/k \\rfloor$ regular $k$-gons for any integer $k\\ge 4$ and that this bound is optimal, and (iv) provide a short proof that the sum of all the distances between its ve","authors_text":"Amol Aggarwal","cross_cats":["cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2010-09-12T05:56:03Z","title":"On Isosceles Triangles and Related Problems in a Convex Polygon"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2218","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:f014b34fe5abe014d4a1b0598412f9faf7d9dc1adf06ece1c90814e94f36d5c3","target":"record","created_at":"2026-05-18T04:40:55Z","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":"d668fbb8741b210d5886a4b8a7ff13652501429ccf8df33d06ece740578de63a","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2010-09-12T05:56:03Z","title_canon_sha256":"a79e2787f61bdbd512b7ca647239d687aadd161cdebc7e82582aa3adebe940e0"},"schema_version":"1.0","source":{"id":"1009.2218","kind":"arxiv","version":2}},"canonical_sha256":"e2d3a2e621e58843c499f098446d72c87d8fc9db416e232f9845bfd0bd014f7e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e2d3a2e621e58843c499f098446d72c87d8fc9db416e232f9845bfd0bd014f7e","first_computed_at":"2026-05-18T04:40:55.728014Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:55.728014Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mpKjjobUn+FZBlK5vVQhpWuE4e1rKd7h2Hcirp+w9g7KG7OeYBOTEe3UualizGUK2R9qwJWDfMMVVSnkzOLTBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:55.728565Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.2218","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f014b34fe5abe014d4a1b0598412f9faf7d9dc1adf06ece1c90814e94f36d5c3","sha256:9150d22b7471afc8475854811fe2f3faa9c0735030ae89284935a45b6493c9a1"],"state_sha256":"cf0bed2a513d0d796946e51964a3e9767002f3602f9d11274ea50902c5a17b1f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rq86d3BoONsR3k3zSPmqx5FfaCaSA5eYeHTnZsNs56+za22CP8Sk4wRo2+4RxDZcOzFioQ/md2qBNZdiDG03CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T01:12:39.810992Z","bundle_sha256":"b9a7ad5ecb4fc27b5726d13dd358e704168a8ea66c93cdce1f8cd6caeae23c31"}}