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This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Carathéodory spaces associated with a family of Lipschitz continuous vector fields. My personal interest in this field goes back to a series of joint papers with E. Lanconelli, where this notion was exploited for the study of pointwise regularity of weak solutions to degenerate elliptic partial differential equations. As stated in the title, here we are mainly concerned...
We construct various Besicovitch sets using Baire category arguments.
We prove some results in the context of isoperimetric inequalities with quantitative terms. In the -dimensional case, our main contribution is a method for determining the optimal coefficients in the inequality , valid for each Borel set with positive and finite area, with and being, respectively, the and the of . In dimensions, besides proving existence and regularity properties of minimizers for a wide class of including the lower semicontinuous extension of , we describe the...
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence...
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