{"paper":{"title":"On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Arnaud Marsiglietti, Artem Zvavitch, Galyna Livshyts, Piotr Nayar","submitted_at":"2015-04-19T19:57:25Z","abstract_excerpt":"In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \\[\n  \\mu(\\lambda A + (1-\\lambda)B)^{1/n} \\geq \\lambda \\mu(A)^{1/n} + (1-\\lambda)\\mu(B)^{1/n} \\] holds true for an unconditional product measure $\\mu$ with decreasing density and a pair of unconditional convex bodies $A,B \\subset \\mathbb{R}^n$. We also show that the above inequality is true for any unconditional $\\log$-concave measure $\\mu$ and unconditional convex bodies $A,B \\subset \\mathbb{R}^n$. Finally, we prove that the inequality is tru"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04878","kind":"arxiv","version":3},"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"}