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From its functional counterpart, for any log-concave random variable $X$, one has $\\mathbb{P}(X\\ge \\mathbb{E}X)\\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Gr\\\"unbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\\mathbb{R}$, then $$\n  h(X)-"},"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":"2607.23269","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-25T16:11:15Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"2e526a3b5884fa6c53d717cbdeda00be56bd8574cda5fbac8866e0a0d5b3a092","abstract_canon_sha256":"afd343174a4381836dfba9e0c2758a65b9e0c9e1d1044942d27851614daf0048"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T01:22:45.085844Z","signature_b64":"lYwOPW0IcfkmJdl153VCCeF3ZqR8XkuT+NqAssCsQtue902scKf5bVN7IcMqYsaeSy7P+75Ji6HK+wj53CZ9AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4893d139e59fe28b64013fc303cfab96fe97ec663bd4ca8cbe21d185a9f14df3","last_reissued_at":"2026-07-28T01:22:45.084978Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T01:22:45.084978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Entropic analogues of Gr\\\"unbaum's inequality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.PR","authors_text":"Lampros Gavalakis, Martin Rapaport, Matthieu Fradelizi","submitted_at":"2026-07-25T16:11:15Z","abstract_excerpt":"The classical Gr\\\"unbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. 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