{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5ZFUVB3NNRV7CFSUENLZ7EIPRB","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":"382658ad4a566c04798fc11faa3cd3804c684e731e8bfc1ad7b0f4be5c68d9b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-05-04T16:39:28Z","title_canon_sha256":"80db4f8bee76219ac0e6ab25c0c33f1e4736041b6d2e4d616e7d9314fe9ac557"},"schema_version":"1.0","source":{"id":"1105.0868","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0868","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0868v2","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0868","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"pith_short_12","alias_value":"5ZFUVB3NNRV7","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"5ZFUVB3NNRV7CFSU","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"5ZFUVB3N","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:2b48e70360f42edd1c817fa505e56913bcf3138e71a45fb71d74a8615277a09a","target":"graph","created_at":"2026-05-18T04:12:56Z","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 $v_1, ..., v_m$ be a finite set of unit vectors in $\\RR^n$. Suppose that an infinite sequence of Steiner symmetrizations are applied to a compact convex set $K$ in $\\RR^n$, where each of the symmetrizations is taken with respect to a direction from among the $v_i$. Then the resulting sequence of Steiner symmetrals always converges, and the limiting body is symmetric under reflection in any of the directions $v_i$ that appear infinitely often in the sequence. In particular, an infinite periodic sequence of Steiner symmetrizations always converges, and the set functional determined by this i","authors_text":"Daniel A. Klain","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-05-04T16:39:28Z","title":"Steiner symmetrization using a finite set of directions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0868","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":"sha256:912f3edd4df2be50655dfe61ae4dc3a2703fb2d7fd0ed1503c09c46fab836888","target":"record","created_at":"2026-05-18T04:12:56Z","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":"382658ad4a566c04798fc11faa3cd3804c684e731e8bfc1ad7b0f4be5c68d9b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-05-04T16:39:28Z","title_canon_sha256":"80db4f8bee76219ac0e6ab25c0c33f1e4736041b6d2e4d616e7d9314fe9ac557"},"schema_version":"1.0","source":{"id":"1105.0868","kind":"arxiv","version":2}},"canonical_sha256":"ee4b4a876d6c6bf1165423579f910f8856cbcdf2b2e40e9b14275c63a7cf7e37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee4b4a876d6c6bf1165423579f910f8856cbcdf2b2e40e9b14275c63a7cf7e37","first_computed_at":"2026-05-18T04:12:56.327345Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:56.327345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TGELOX0cTTuOx5ApqndJgXAKiKR8pWoXfstFvPtRCBayLMQJaC/DJ6MiWDgb/BB44tVBDOGeXkJFuyd0u0BzAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:56.327829Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.0868","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:912f3edd4df2be50655dfe61ae4dc3a2703fb2d7fd0ed1503c09c46fab836888","sha256:2b48e70360f42edd1c817fa505e56913bcf3138e71a45fb71d74a8615277a09a"],"state_sha256":"d9a65818760b1d742993bdfaa38465e858c20aeb3ae6b5d912bab5afb31f84b7"}