{"paper":{"title":"An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Katalin Marton","submitted_at":"2012-06-21T13:24:55Z","abstract_excerpt":"Let $q(x)$ and $p(x)$ denote density functions on the $n$-dimensional Euclidean space, and let $p_i(\\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\\cdot|x_1,..., x_{i-1},x_{i+1},..., x_n)$ denote their local specifications. For a class of density functions $q$ we prove an inequality between the relative entropy $D(p||q)$ and a weighted sum of the conditional relative entropies $D(p_i(\\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n) ||Q_i(\\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n))$ that holds for any $p$. The weights are proportional to the logarithmic Sobolev constants of the local specifications o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4868","kind":"arxiv","version":1},"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"}