{"paper":{"title":"Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexander V. Kolesnikov","submitted_at":"2012-01-11T16:26:42Z","abstract_excerpt":"We study the optimal transportation mapping $\\nabla \\Phi : \\mathbb{R}^d \\mapsto \\mathbb{R}^d$ pushing forward a probability measure $\\mu = e^{-V} \\ dx$ onto another probability measure $\\nu = e^{-W} \\ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \\Phi$ on $\\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\\mathbb{R}^d, g, \\mu)$.\n  We prove, in particular, that $M$ admits a non-negative Bakry--{\\'E}mery tensor provided both $V$ and $W$ are convex. If the target measure $\\nu$ is the Lebesgue measure on a convex set $\\Omega$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2342","kind":"arxiv","version":3},"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"}