Tubular neighborhoods in the sub-Riemannian Heisenberg groups

We consider the Carnot-Carath'eodory distance ${\delta }_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups ${\mathbb{H}}^n$, $n\ge 1$. The $\mathbb{H}$-regularity of ${\delta }_E$ is proved under mild conditions involving a general notion of singular points. In case $E$ is a Euclidean $C^k$ submanifold, $k\ge 2$, we prove that ${\delta }_E$ is $C^k$ out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of $E$ is of class $C^2$ are obtained, out of the singular set, in terms of the horizontal principal curvatures of $\mathrm{\partial }E$ and of the function $⟨N,T⟩\mathrm{/}\mathrm{\mid }{N}_h\mathrm{\mid }$ and its tangent derivatives.