{"paper":{"title":"Measure contraction properties of Carnot groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.OC"],"primary_cat":"math.MG","authors_text":"Luca Rizzi","submitted_at":"2015-10-20T16:56:26Z","abstract_excerpt":"We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\\mathrm{MCP}(K,N)$ if and only if $K \\leq 0$ and $N \\geq k+3$. This generalizes the well known result by Juillet for the Heisenberg group $\\mathbb{H}_{k+1}$ to a larger class of structures, which admit non-trivial abnormal minimizing curves.\n  The number $k+3$ coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least $N$ such that the $\\mathrm{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05960","kind":"arxiv","version":4},"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"}