{"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; the key supporting estimate is ∫ |∇ψ| dμ ≤ C n for isotropic log-concave μ with density e^{-ψ}.","one_line_summary":"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).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"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.","pith_extraction_headline":"Even log-concave measures satisfy a Brunn-Minkowski inequality for convex sets with exponent at least c over n cubed log n."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.02747/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T15:34:26.189763Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T02:31:22.432062Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:02:05.165959Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b142f1940090ddb9228f93b923454e7b57f801f856926566e725bc95cf855764"},"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"}