Pansu–Wulff shapes in ℍ¹

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing origin its interior. Associated to there is perimeter functional, that coincides with classical sub-Riemannian case K closed unit disk centered at of {{\mathbb{R}}^{2}} . Under assumption has C {C^{2}} boundary strictly positive geodesic curvature we compute variation formula for sets boundary. The localization variational non-singular part boundary, composed points where tangent plane not horizontal, allows us define mean function H {H_{K}} out singular set. In non-vanishing curvature, condition be constant implies portion foliated horizontal liftings translations ∂ ⁡ {\partial K} dilated factor {\frac{1}{H_{K}}} Based on this can sphere