{"paper":{"title":"Remarks on curvature in the transportation metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alexander Kolesnikov, Bo'az Klartag","submitted_at":"2016-04-14T14:18:43Z","abstract_excerpt":"According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the \"hyperbolic\" toric K\\\"ahler-Einstein equation $e^{\\Phi} = \\det D^2 \\Phi$ on proper convex cones. We prove a generalization of this theorem by showing that for every\n  $\\Phi$ solving this equation on a proper convex domain $\\Omega$ the corresponding metric measure space $(D^2 \\Phi, e^{\\Phi}dx)$ has a non-positive Bakry-{\\'E}mery tensor. Modifying the Calabi computatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.04165","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"}