{"paper":{"title":"Comparison results for capacity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ana Hurtado, Manuel Ritor\\'e, Vicente Palmer","submitted_at":"2010-12-02T16:49:24Z","abstract_excerpt":"We obtain in this paper bounds for the capacity of a compact set $K$. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold, has smooth boundary, and the principal curvatures of $\\partial K$ are larger than or equal to $H_0>0$, then ${\\rm Cap}(K)\\geq (n-1)\\,H_0{\\rm vol}(\\partial K)$. When $K$ is contained in an $(n+1)$-dimensional manifold with non-negative Ricci curvature, has smooth boundary, and the mean curvature of $\\partial K$ is smaller than or equal to $H_0$, we prove the inequality ${\\rm Cap}(K)\\leq (n-1)\\,H_0{\\rm vol}(\\partial K)$. In both cases we are able to charac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0487","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"}