{"paper":{"title":"Entropy functions and determinant inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Dongning Guo, Raymond Yeung, Terence Chan","submitted_at":"2012-01-25T11:51:34Z","abstract_excerpt":"In this paper, we show that the characterisation of all determinant inequalities for $n \\times n$ positive definite matrices is equivalent to determining the smallest closed and convex cone containing all entropy functions induced by $n$ scalar Gaussian random variables. We have obtained inner and outer bounds on the cone by using representable functions and entropic functions. In particular, these bounds are tight and explicit for $n \\le 3$, implying that determinant inequalities for $3 \\times 3$ positive definite matrices are completely characterized by Shannon-type information inequalities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5241","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"}