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Then $$ \\frac{1}{|\\Omega|} \\int_{\\Omega}{f dx} \\leq \\frac{c_n}{ |\\partial \\Omega| } \\int_{\\partial \\Omega}{ f d\\sigma},$$ where $c_n \\leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \\geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ \\Omega_2 \\subset \\Omega_1"},"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":"1907.06122","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-13T20:16:39Z","cross_cats_sorted":["math.FA","math.MG"],"title_canon_sha256":"19bc2a287987b8e100feccbaa18607c4c38f374bdef78b67e37ffdeacd0d5382","abstract_canon_sha256":"cd1ff88bf35c027fe09450c227af8e255cdadf95977203d9635429d7f50f0b7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:42.373679Z","signature_b64":"SNV4XFkcxMlnriCWt5BBhfhQ16i3dtn+C8QzMiWUROdJnFgBriWWGHaF8hE4gPMyTWCDpMXka5oAlNkvggEPDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"562be807c61ae9eac9ada5013ca64cc7c08f4ccebd20f60f4acfcc1c70eada0c","last_reissued_at":"2026-05-17T23:40:42.373109Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:42.373109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.MG"],"primary_cat":"math.CA","authors_text":"Antoine Henrot, Barbara Brandolini, Jeffrey J. 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