{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:B52QECA3DXGMXVDWSNZGZ3NHFF","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":"e2eb1d5d4fa9989197ad544285e2cfc36b2dc3e395f6c9340164d8212dca4c51","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-05-19T13:35:37Z","title_canon_sha256":"c58958817491edda5352d4195e3ef445fe2c2774e325a6e6218f26c7a5d83134"},"schema_version":"1.0","source":{"id":"1205.4327","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.4327","created_at":"2026-05-18T03:36:04Z"},{"alias_kind":"arxiv_version","alias_value":"1205.4327v2","created_at":"2026-05-18T03:36:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4327","created_at":"2026-05-18T03:36:04Z"},{"alias_kind":"pith_short_12","alias_value":"B52QECA3DXGM","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"B52QECA3DXGMXVDW","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"B52QECA3","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:22dc7d004274afbff2510774307af11793ec938c9cda148c2901f9f4148c39e9","target":"graph","created_at":"2026-05-18T03:36:04Z","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":"An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$-dimensional Euclidean space $\\R^n$. For example, it is proved that if $n\\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, and associative if and only if it is $L_p$ addition for some $1\\le p\\le\\infty$. It is also demonstrated that if $n\\ge 2$, an operation * between compact convex set","authors_text":"Daniel Hug, Richard J. Gardner, Wolfgang Weil","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-05-19T13:35:37Z","title":"Operations between sets in geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4327","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:dd90e3045413510f1464e0c0b728d202043f3d72ebc679121df08069969bf9c9","target":"record","created_at":"2026-05-18T03:36:04Z","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":"e2eb1d5d4fa9989197ad544285e2cfc36b2dc3e395f6c9340164d8212dca4c51","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-05-19T13:35:37Z","title_canon_sha256":"c58958817491edda5352d4195e3ef445fe2c2774e325a6e6218f26c7a5d83134"},"schema_version":"1.0","source":{"id":"1205.4327","kind":"arxiv","version":2}},"canonical_sha256":"0f7502081b1dcccbd47693726ceda7296b62fb7d3c3f036d83f716c058a7a1be","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f7502081b1dcccbd47693726ceda7296b62fb7d3c3f036d83f716c058a7a1be","first_computed_at":"2026-05-18T03:36:04.545625Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:36:04.545625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UzLFJ4nvfCi1aA1Nhn6+7w8uuQIWAR5yr3+nIr/3yRt9tcprRsK5oYJwjhUtUr0El8OYmjGR7IFYkZshC/IzBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:36:04.546112Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.4327","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd90e3045413510f1464e0c0b728d202043f3d72ebc679121df08069969bf9c9","sha256:22dc7d004274afbff2510774307af11793ec938c9cda148c2901f9f4148c39e9"],"state_sha256":"0881614b62e484d623110c064def58cabe81fea57ca9e9e5d5fd89d14949ed5d"}