{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:JZ7V73OCWD22CBG3I6WLMPHXXK","short_pith_number":"pith:JZ7V73OC","canonical_record":{"source":{"id":"1012.3610","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-16T14:13:09Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"ca02b61afbc04f9685cd5f00bfb4eede715f8808683e15952d0a3d5e60ec69f1","abstract_canon_sha256":"6f8f3c8d522d3ab8cfff69f3a67988ef0fd25f672e6718833b99d7a02b8ea07d"},"schema_version":"1.0"},"canonical_sha256":"4e7f5fedc2b0f5a104db47acb63cf7ba864c4534b710b3d394dd4b83283c72fa","source":{"kind":"arxiv","id":"1012.3610","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.3610","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"arxiv_version","alias_value":"1012.3610v1","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3610","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"pith_short_12","alias_value":"JZ7V73OCWD22","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JZ7V73OCWD22CBG3","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JZ7V73OC","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:JZ7V73OCWD22CBG3I6WLMPHXXK","target":"record","payload":{"canonical_record":{"source":{"id":"1012.3610","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-16T14:13:09Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"ca02b61afbc04f9685cd5f00bfb4eede715f8808683e15952d0a3d5e60ec69f1","abstract_canon_sha256":"6f8f3c8d522d3ab8cfff69f3a67988ef0fd25f672e6718833b99d7a02b8ea07d"},"schema_version":"1.0"},"canonical_sha256":"4e7f5fedc2b0f5a104db47acb63cf7ba864c4534b710b3d394dd4b83283c72fa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:02.428104Z","signature_b64":"wzs2YZtcc361xX4YALNnchc3wfHheKj7yaDRHOOjt5dsXTzk9MpN2DWlIMCtyb33juJ3y3fbwOdgojaniLpKCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e7f5fedc2b0f5a104db47acb63cf7ba864c4534b710b3d394dd4b83283c72fa","last_reissued_at":"2026-05-18T03:07:02.427470Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:02.427470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.3610","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RilOjQ1scBzlErrpBKnWm7FD+iYRbzEOG+kzsCV9GC7ypmYdB95ncM+HMjx9Y/zWRT06rbAiR9E+5xRg8NAzAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T11:19:36.068661Z"},"content_sha256":"a20247510aca611cb1b9cc380f614b279337faee325e469126ab110ae8f7f1ed","schema_version":"1.0","event_id":"sha256:a20247510aca611cb1b9cc380f614b279337faee325e469126ab110ae8f7f1ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:JZ7V73OCWD22CBG3I6WLMPHXXK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inverse Additive Problems for Minkowski Sumsets II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"D. J. Grynkiewicz, G. A. Freiman, O. Serra, Y. Stanchescu","submitted_at":"2010-12-16T14:13:09Z","abstract_excerpt":"The Brunn-Minkowski Theorem asserts that $\\mu_d(A+B)^{1/d}\\geq \\mu_d(A)^{1/d}+\\mu_d(B)^{1/d}$ for convex bodies $A,\\,B\\subseteq \\R^d$, where $\\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and only if $A$ and $B$ are homothetic, but few characterizations of equality in other related bounds are known. Let $H$ be a hyperplane. Bonnesen later strengthened this bound by showing $$\\mu_d(A+B)\\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\\frac{\\mu_d(A)}{M}+\\frac{\\mu_d(B)}{N}),$$ where $M=\\sup\\{\\mu_{d-1}((\\mathbf x+H)\\cap A)\\mid \\mathbf x\\in \\R^d\\}$ and $N=\\sup\\{\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3610","kind":"arxiv","version":1},"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":"2026-05-18T03:07:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VLk24EBa4xBz+p71HTTjgyUjqnZarm7jwCgqEL1l5dMNbuFluOKT9UKYex0DTEz3tWGKeoYgnidtrzYjuGbMAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T11:19:36.069012Z"},"content_sha256":"1c503bf5ceef9acb76eaa545d97770ce85bda01e65ccf35552a0a7a914ceded8","schema_version":"1.0","event_id":"sha256:1c503bf5ceef9acb76eaa545d97770ce85bda01e65ccf35552a0a7a914ceded8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JZ7V73OCWD22CBG3I6WLMPHXXK/bundle.json","state_url":"https://pith.science/pith/JZ7V73OCWD22CBG3I6WLMPHXXK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JZ7V73OCWD22CBG3I6WLMPHXXK/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":"2026-06-26T11:19:36Z","links":{"resolver":"https://pith.science/pith/JZ7V73OCWD22CBG3I6WLMPHXXK","bundle":"https://pith.science/pith/JZ7V73OCWD22CBG3I6WLMPHXXK/bundle.json","state":"https://pith.science/pith/JZ7V73OCWD22CBG3I6WLMPHXXK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JZ7V73OCWD22CBG3I6WLMPHXXK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:JZ7V73OCWD22CBG3I6WLMPHXXK","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":"6f8f3c8d522d3ab8cfff69f3a67988ef0fd25f672e6718833b99d7a02b8ea07d","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-16T14:13:09Z","title_canon_sha256":"ca02b61afbc04f9685cd5f00bfb4eede715f8808683e15952d0a3d5e60ec69f1"},"schema_version":"1.0","source":{"id":"1012.3610","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.3610","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"arxiv_version","alias_value":"1012.3610v1","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3610","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"pith_short_12","alias_value":"JZ7V73OCWD22","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JZ7V73OCWD22CBG3","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JZ7V73OC","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:1c503bf5ceef9acb76eaa545d97770ce85bda01e65ccf35552a0a7a914ceded8","target":"graph","created_at":"2026-05-18T03:07:02Z","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":"The Brunn-Minkowski Theorem asserts that $\\mu_d(A+B)^{1/d}\\geq \\mu_d(A)^{1/d}+\\mu_d(B)^{1/d}$ for convex bodies $A,\\,B\\subseteq \\R^d$, where $\\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and only if $A$ and $B$ are homothetic, but few characterizations of equality in other related bounds are known. Let $H$ be a hyperplane. Bonnesen later strengthened this bound by showing $$\\mu_d(A+B)\\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\\frac{\\mu_d(A)}{M}+\\frac{\\mu_d(B)}{N}),$$ where $M=\\sup\\{\\mu_{d-1}((\\mathbf x+H)\\cap A)\\mid \\mathbf x\\in \\R^d\\}$ and $N=\\sup\\{\\m","authors_text":"D. J. Grynkiewicz, G. A. Freiman, O. Serra, Y. Stanchescu","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-16T14:13:09Z","title":"Inverse Additive Problems for Minkowski Sumsets II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3610","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:a20247510aca611cb1b9cc380f614b279337faee325e469126ab110ae8f7f1ed","target":"record","created_at":"2026-05-18T03:07:02Z","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":"6f8f3c8d522d3ab8cfff69f3a67988ef0fd25f672e6718833b99d7a02b8ea07d","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-16T14:13:09Z","title_canon_sha256":"ca02b61afbc04f9685cd5f00bfb4eede715f8808683e15952d0a3d5e60ec69f1"},"schema_version":"1.0","source":{"id":"1012.3610","kind":"arxiv","version":1}},"canonical_sha256":"4e7f5fedc2b0f5a104db47acb63cf7ba864c4534b710b3d394dd4b83283c72fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e7f5fedc2b0f5a104db47acb63cf7ba864c4534b710b3d394dd4b83283c72fa","first_computed_at":"2026-05-18T03:07:02.427470Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:02.427470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wzs2YZtcc361xX4YALNnchc3wfHheKj7yaDRHOOjt5dsXTzk9MpN2DWlIMCtyb33juJ3y3fbwOdgojaniLpKCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:02.428104Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.3610","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a20247510aca611cb1b9cc380f614b279337faee325e469126ab110ae8f7f1ed","sha256:1c503bf5ceef9acb76eaa545d97770ce85bda01e65ccf35552a0a7a914ceded8"],"state_sha256":"9da9d6b5980386719363ec43ded76ae4a3232ba07d79aaebfe4f454c15e7e7be"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KKGwZqjT2w1MB1kizB0Ouv0Ve0sm4h1gqWEiyoBurmbBjcPMmg3hPrRS7byjpd9lr+Ik4noOKphtTTudtu3/DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T11:19:36.070999Z","bundle_sha256":"18652258cf12a76031543827b62a530a92207b77cbc558a966df708493c5198c"}}