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First, we study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\\mu$ on $\\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. We prove that for every pair of symmetric convex sets $K,L$ in $\\mathbb{R}^n$ and every $\\lambda\\in(0,1)$, $$\\mu(\\lambda K+(1-\\lambda)L)^{c_n} \\geq \\lambda \\mu(K)^{c_n}+(1-\\lambda)\\mu(L)^{c_n},$$ where $c_n\\geq c/n^3\\ln n$ for some absolute constant $c>0$. Secondly, we study the maximal perim"},"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":"2605.02747","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-05-04T15:48:01Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"4624e96bc1bd2b085866fd0fa18a9186f340476df844142dd7cc4e145ab2a0e2","abstract_canon_sha256":"01423a287a013ec23ab99d946b134b69ba12e1f00c97915264729f352c4e834a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T01:09:50.360393Z","signature_b64":"iJ37h34x8ZVAoTN4rzFs9L2qeyDHExxNCAMqYa5X7PG1uFDuKsTctzNo1sAFXN5rYlOBNBA727HgWPvvHLXJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"085390455902f243861398afcea2e3a56be968427a8172a943bfc31e2036b0b3","last_reissued_at":"2026-06-04T01:09:50.359540Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T01:09:50.359540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Even log-concave measures satisfy a Brunn-Minkowski inequality for convex sets with exponent at least c over n cubed log n.","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Alexandros Eskenazis, Apostolos Giannopoulos, Natalia Tziotziou","submitted_at":"2026-05-04T15:48:01Z","abstract_excerpt":"This paper is dedicated to two geometric problems associated to log-concave measures on $\\mathbb{R}^n$. First, we study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\\mu$ on $\\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. We prove that for every pair of symmetric convex sets $K,L$ in $\\mathbb{R}^n$ and every $\\lambda\\in(0,1)$, $$\\mu(\\lambda K+(1-\\lambda)L)^{c_n} \\geq \\lambda \\mu(K)^{c_n}+(1-\\lambda)\\mu(L)^{c_n},$$ where $c_n\\geq c/n^3\\ln n$ for some absolute constant $c>0$. Secondly, we study the maximal perim"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every pair of symmetric convex sets K, L in R^n and every λ in (0,1), μ(λK + (1-λ)L)^{c_n} ≥ λ μ(K)^{c_n} + (1-λ) μ(L)^{c_n} where c_n ≥ c / (n^3 ln n) for some absolute c > 0; the key supporting estimate is ∫ |∇ψ| dμ ≤ C n for isotropic log-concave μ with density e^{-ψ}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The measures are even and log-concave (so the density is e^{-ψ} with ψ convex), and the gradient integral bound holds for the isotropic case; if this integral bound fails or the evenness is dropped, the claimed exponent on the Brunn-Minkowski inequality need not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Even log-concave measures satisfy μ(λK + (1-λ)L)^{c_n} ≥ λ μ(K)^{c_n} + (1-λ) μ(L)^{c_n} with c_n ≥ c/(n^3 ln n).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Even log-concave measures satisfy a Brunn-Minkowski inequality for convex sets with exponent at least c over n cubed log n.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"69a50d1cff5ef7c8bf115bef2d009f2ff91528ec6d285e95f0ba0c4bfefe272e"},"source":{"id":"2605.02747","kind":"arxiv","version":2},"verdict":{"id":"2dd1f240-2a74-43ae-9f88-bfe4678b1961","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T01:50:00.554495Z","strongest_claim":"For every pair of symmetric convex sets K, L in R^n and every λ in (0,1), μ(λK + (1-λ)L)^{c_n} ≥ λ μ(K)^{c_n} + (1-λ) μ(L)^{c_n} where c_n ≥ c / (n^3 ln n) for some absolute c > 0; 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