{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:1995:6X2TLLCUHH6CBEZCPSOPDRB6DW","short_pith_number":"pith:6X2TLLCU","schema_version":"1.0","canonical_sha256":"f5f535ac5439fc2093227c9cf1c43e1dbb6d091152944a1379857ff916a2ff4c","source":{"kind":"arxiv","id":"math/9503214","version":1},"attestation_state":"computed","paper":{"title":"Lattice coverings and gaussian measures of n-dimensional convex bodies","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Stanislaw J. Szarek, W. Banaszczyk","submitted_at":"1995-03-20T00:00:00Z","abstract_excerpt":"Let $\\| \\cdot \\|$ be the euclidean norm on ${\\bf R}^n$ and $\\gamma_n$ the (standard) Gaussian measure on ${\\bf R}^n$ with density $(2 \\pi )^{-n/2} e^{- \\| x\\|^2 /2}$. Let $\\vartheta$ ($ \\simeq 1.3489795$) be defined by $\\gamma_1 ([ - \\vartheta /2, \\vartheta /2]) = 1/2$ and let $L$ be a lattice in ${\\bf R}^n$ generated by vectors of norm $\\leq \\vartheta$. Then, for any closed convex set $V$ in ${\\bf R}^n$ with $\\gamma_n (V) \\geq \\frac{1}{2}$ and for any $a \\in {\\bf R}^n$, $(a +L) \\cap V \\neq \\phi$. The above statement can be viewed as a ``nonsymmetric'' version of Minkowski Theorem."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/9503214","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"1995-03-20T00:00:00Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"8d9238edc4f9d4859f2967085cd1c2bb25ac3ce671e6641902ef74e3f0862ec6","abstract_canon_sha256":"e3e1ea37e054da79f6d1e8f90bdd536f1e1bec89a28aace320bed9c5813bd722"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:50.383827Z","signature_b64":"0cAgTYhpvlR5x3cKteOpwPaRRGc3/L90bHOGEPVwCuokZR8daIhCeDz5TFWU5sLVNeRqaJxZAPr/pyIp7pQzBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5f535ac5439fc2093227c9cf1c43e1dbb6d091152944a1379857ff916a2ff4c","last_reissued_at":"2026-05-18T01:05:50.383427Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:50.383427Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lattice coverings and gaussian measures of n-dimensional convex bodies","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Stanislaw J. Szarek, W. Banaszczyk","submitted_at":"1995-03-20T00:00:00Z","abstract_excerpt":"Let $\\| \\cdot \\|$ be the euclidean norm on ${\\bf R}^n$ and $\\gamma_n$ the (standard) Gaussian measure on ${\\bf R}^n$ with density $(2 \\pi )^{-n/2} e^{- \\| x\\|^2 /2}$. Let $\\vartheta$ ($ \\simeq 1.3489795$) be defined by $\\gamma_1 ([ - \\vartheta /2, \\vartheta /2]) = 1/2$ and let $L$ be a lattice in ${\\bf R}^n$ generated by vectors of norm $\\leq \\vartheta$. Then, for any closed convex set $V$ in ${\\bf R}^n$ with $\\gamma_n (V) \\geq \\frac{1}{2}$ and for any $a \\in {\\bf R}^n$, $(a +L) \\cap V \\neq \\phi$. The above statement can be viewed as a ``nonsymmetric'' version of Minkowski Theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9503214","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/9503214","created_at":"2026-05-18T01:05:50.383485+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9503214v1","created_at":"2026-05-18T01:05:50.383485+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9503214","created_at":"2026-05-18T01:05:50.383485+00:00"},{"alias_kind":"pith_short_12","alias_value":"6X2TLLCUHH6C","created_at":"2026-05-18T12:25:47.700082+00:00"},{"alias_kind":"pith_short_16","alias_value":"6X2TLLCUHH6CBEZC","created_at":"2026-05-18T12:25:47.700082+00:00"},{"alias_kind":"pith_short_8","alias_value":"6X2TLLCU","created_at":"2026-05-18T12:25:47.700082+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW","json":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW.json","graph_json":"https://pith.science/api/pith-number/6X2TLLCUHH6CBEZCPSOPDRB6DW/graph.json","events_json":"https://pith.science/api/pith-number/6X2TLLCUHH6CBEZCPSOPDRB6DW/events.json","paper":"https://pith.science/paper/6X2TLLCU"},"agent_actions":{"view_html":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW","download_json":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW.json","view_paper":"https://pith.science/paper/6X2TLLCU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9503214&json=true","fetch_graph":"https://pith.science/api/pith-number/6X2TLLCUHH6CBEZCPSOPDRB6DW/graph.json","fetch_events":"https://pith.science/api/pith-number/6X2TLLCUHH6CBEZCPSOPDRB6DW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW/action/storage_attestation","attest_author":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW/action/author_attestation","sign_citation":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW/action/citation_signature","submit_replication":"https://pith.science/pith/6X2TLLCUHH6CBEZCPSOPDRB6DW/action/replication_record"}},"created_at":"2026-05-18T01:05:50.383485+00:00","updated_at":"2026-05-18T01:05:50.383485+00:00"}