Geometric Hardy inequalities on the Heisenberg groups via convexity

We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carath\'eodory distance on half-spaces and arbitrary bounded convex domains of ${\mathbb{H}}^n$. In all cases the constant $((p-1)/p)^p$ is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak $H$-concavity of the Euclidean distance to the boundary, thus obtaining a proof of the $L^p$-Hardy inequality on convex domains.