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 $\partial E$ and of the function $\langle N,T\rangle/|N_h|$ and its tangent derivatives.
Categories: math.MG math.DG