Steiner's formula in the Heisenberg group (Q494200)

Steiner's formula states that the volume of an \(\epsilon\)-neighborhood of a smooth bounded regular domain in \(\mathbb{R}^n\) is a polynomial of degree \(n\) in the variable \(\epsilon\) whose coefficients are curvature integrals. The authors prove a similar result in the sub-Riemannian setting of the first Heisenberg group \(\mathbb{H}\). In contrast to the Euclidean setting, they find that the volume of a (localized) \(\epsilon\)-neighborhood of a smooth bounded regular domain \(\Omega\) with respect to the Heisenberg metric in \(\mathbb{H}\) is generally not a polynomial, but locally it is an analytic function of \(\epsilon\) having a power series expansion whose coefficients can be explicitly written in terms of integrals of iterated divergences of the signed Carnot-Carathédory distance function from the smooth regular boundary \(\partial\Omega\) of \(\Omega\).