2016/07/15 by Andrea Pinamonti, Pinamonti, Andrea, Gareth Speight +1
Mathematics · #28A75 #43A80 #49Q15 #53C17 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1607.04681
openalex publication_date 2016/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not σ-porous with respect to the Carnot-Carathéodory (CC) distance. In the first Heisenberg group we observe that there exist sets which are porous with respect to the CC distance but not the Euclidean distance and vice-versa. In Carnot groups we then construct a Lipschitz function which is Pansu differentiable at no point of a given σ-porous set and show preimages of open sets under the horizontal gradient are far from being porous.