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• W3098495011 abstract "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 $$K le 0$$ and $$N ge k+3$$ . 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. 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 $$mathrm {MCP}(0,N)$$ is satisfied]. We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. When applied to Carnot groups, our results improve a previous lower bound due to Rifford. As a byproduct, we prove that a Carnot group is ideal if and only if it is fat." @default.
• W3098495011 created "2020-11-23" @default.
• W3098495011 creator A5039140022 @default.
• W3098495011 date "2016-05-18" @default.
• W3098495011 modified "2023-09-30" @default.
• W3098495011 title "Measure contraction properties of Carnot groups" @default.
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• W3098495011 doi "https://doi.org/10.1007/s00526-016-1002-y" @default.
• W3098495011 hasPublicationYear "2016" @default.
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