{"paper":{"title":"Some remarks about the maximal perimeter of convex sets with respect to probability measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.PR"],"primary_cat":"math.MG","authors_text":"Galyna V. Livshyts","submitted_at":"2019-04-15T02:48:32Z","abstract_excerpt":"In this note we study the maximal perimeter of a convex set in $\\mathbb{R}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $\\mu$ on $ \\mathbb{R}^n$, satisfying very mild assumptions, there exists a convex set of $\\mu$-perimeter at least $C\\frac{\\sqrt{n}}{\\sqrt[4]{Var|X|} \\sqrt{\\mathbb{E}|X|}}.$ This implies, in particular, that for any isotropic log-concave measure $\\mu$ one may find a convex set of $\\mu$- perimeter of order $n^{\\frac{1}{8}}$.\n  Secondly, we derive a general upper bound of $Cn|| f||^{\\frac{1}{n}}_{\\infty}$ on the maximal perimete"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06814","kind":"arxiv","version":2},"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"}