Steiner’s formula in the Heisenberg group

Steiner’s tube formula states that the volume of an ϵϵ-neighborhood of a smooth regular domain in RnRn is a polynomial of degree nn in the variable ϵϵ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ϵϵ-neighborhood with respect to the Heisenberg metric is an analytic function of ϵϵ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms.