{"paper":{"title":"Riemannian metrics on convex sets with applications to Poincar\\'e and log-Sobolev inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.SP"],"primary_cat":"math.FA","authors_text":"Alexander V. Kolesnikov, Emanuel Milman","submitted_at":"2015-10-10T19:16:17Z","abstract_excerpt":"Given a probability measure $\\mu$ supported on a convex subset $\\Omega$ of Euclidean space $(\\mathbb{R}^d,g_0)$, we are interested in obtaining Poincar\\'e and log-Sobolev type inequalities on $(\\Omega,g_0,\\mu)$. To this end, we change the metric $g_0$ to a more general Riemannian one $g$, adapted in a certain sense to $\\mu$, and perform our analysis on $(\\Omega,g,\\mu)$. The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when $\\mu$ is unconditional, i.e. invariant under reflection with respec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02971","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"}