{"paper":{"title":"An Isoperimetric Problem With Density and the Hardy Sobolev Inequality in $\\mathbb{R}^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gyula Csat\\'o","submitted_at":"2014-10-29T16:05:46Z","abstract_excerpt":"We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\\in\\re$, $\\Omega\\subset\\re^2$ then the inequality $$\n  \\left(\\frac{|\\Omega|}{\\pi}\\right)^{\\frac{p+1}{2}}\\leq\\frac{1}{2\\pi}\\int_{\\delomega}|x|^pd\\sigma(x) $$ holds true under appropriate assumptions on $\\Omega$ and $p.$ This solves an open problem arising in the context of isoperimetric problems with density and poses some new ones (for instance generalizations to $\\re^n$). We prove the equivalence with a Hardy-Sobolev inequality, giving the best constant, and generalize th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8041","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"}