{"paper":{"title":"p-capacity vs surface-area","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jie Xiao","submitted_at":"2015-06-11T20:26:41Z","abstract_excerpt":"This paper is devoted to exploring the relationship between the $[1,n)\\ni p$-capacity and the surface-area in $\\mathbb R^{n\\ge 2}$ which especially shows: if $\\Omega\\subset\\mathbb R^n$ is a convex, compact, smooth set with its interior $\\Omega^\\circ\\not=\\emptyset$ and the mean curvature $H(\\partial\\Omega,\\cdot)>0$ of its boundary $\\partial\\Omega$ then $$ \\left(\\frac{n(p-1)}{p(n-1)}\\right)^{p-1}\\le\\frac{\\left(\\frac{\\hbox{cap}_p(\\Omega)}{\\big(\\frac{p-1}{n-p}\\big)^{1-p}\\sigma_{n-1}}\\right)}{\\left(\\frac{\\hbox{area}(\\partial\\Omega)}{\\sigma_{n-1}}\\right)^\\frac{n-p}{n-1}}\\le\\left(\\sqrt[n-1]{\\int_{\\pa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03827","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"}