{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:EPOEBVZGDQUGMH66Y7FAO4Z4GC","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":"8a8777f9297b2c500c3e99e61e09a155d90e4030783b816887b075c7e2b73abb","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-16T13:45:56Z","title_canon_sha256":"05b36a983115f2c9dcbca0e05e74f729940f082718da85d4bc14325efddcc9d3"},"schema_version":"1.0","source":{"id":"1007.2775","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2775","created_at":"2026-05-18T04:14:46Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2775v1","created_at":"2026-05-18T04:14:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2775","created_at":"2026-05-18T04:14:46Z"},{"alias_kind":"pith_short_12","alias_value":"EPOEBVZGDQUG","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EPOEBVZGDQUGMH66","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EPOEBVZG","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:6a730055015df417a367f13591d4506380b671835033025be011c0cf8dc529dc","target":"graph","created_at":"2026-05-18T04:14:46Z","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":"Let $E_d(n)$ be the maximum number of pairs that can be selected from a set of $n$ points in $R^d$ such that the midpoints of these pairs are convexly independent. We show that $E_2(n)\\geq \\Omega(n\\sqrt{\\log n})$, which answers a question of Eisenbrand, Pach, Rothvo\\ss, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also show that $\\lfloor\\frac{1}{3}n^2\\rfloor\\leq E_3(n)\\leq \\frac{3}{8}n^2+O(n^{3/2})$.\n  Let $W_d(n)$ be the maximum number of pairwise nonparallel unit distance pairs i","authors_text":"Konrad J. Swanepoel, Pavel Valtr","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-16T13:45:56Z","title":"Large convexly independent subsets of Minkowski sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2775","kind":"arxiv","version":1},"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:c29e8c79d61e01587859b2b0f87d49ced63d2084a6123dabd06757d37ec1d590","target":"record","created_at":"2026-05-18T04:14:46Z","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":"8a8777f9297b2c500c3e99e61e09a155d90e4030783b816887b075c7e2b73abb","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-16T13:45:56Z","title_canon_sha256":"05b36a983115f2c9dcbca0e05e74f729940f082718da85d4bc14325efddcc9d3"},"schema_version":"1.0","source":{"id":"1007.2775","kind":"arxiv","version":1}},"canonical_sha256":"23dc40d7261c28661fdec7ca07733c30acfbaac4b1b02f83039dca019df1e2c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"23dc40d7261c28661fdec7ca07733c30acfbaac4b1b02f83039dca019df1e2c7","first_computed_at":"2026-05-18T04:14:46.169468Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:46.169468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vWgc9gFP1zrRZ1Zj7eVae9h0324nYnTQSx7pBsGToWLSKap6XdfZxbtSAz91hGA8fxEPxMaLJVQtIRIVbXDlDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:46.170080Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.2775","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c29e8c79d61e01587859b2b0f87d49ced63d2084a6123dabd06757d37ec1d590","sha256:6a730055015df417a367f13591d4506380b671835033025be011c0cf8dc529dc"],"state_sha256":"16b275a8641ae4e4082cff579aeb155ba0918416e0e128d4a7c46cad0a92e6bf"}