Notions of Convexity in Carnot Groups

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Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, background): “We note that, for surfaces in a Carnot group, this same notion of curvature was also already explicitly introduced in [24].”
Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group (2016, background): “A different notion of intrinsic Gaussian curvature for graphs has been suggested in [18].”
Geodetically Convex Sets in the Heisenberg Group (2005, background): “A weak and a strong definition of convex function are discussed in [4]: roughly speaking, the function is required to be convex along the integral curves of the left invariant horizontal vector fields.”
Hardy-type inequalities related to degenerate elliptic differential operators (2006, methodology): “For H-convex function on Carnot groups, we refer the interested reader to [22, 38, 39].”
Constant mean curvature surfaces in sub-Riemannian geometry (2005, methodology): “For example, Danielli, Garofalo and Nhieu ([7],[8]) use a symmetrized horizontal Hessian”
Convex functions on Carnot groups. (2007, background): “Condition i) is called horizontal convexity in [13] and H-convexity in [7].”