Explicit Formulae for Geodesics in Left–Invariant Sub–Finsler Problems on Heisenberg Groups via Convex Trigonometry

Abstract

In the present paper, we obtain explicit formulae for geodesics in some left–invariant sub–Finsler problems on Heisenberg groups $\mathbb {H}_{2n+1}$ . Our main assumption is the following: the compact convex set of unit velocities at identity admits a generalization of spherical coordinates. This includes convex hulls and sums of coordinate 2–dimensional sets, all left–invariant sub–Riemannian structures on $\mathbb {H}_{2n+1}$ , and unit balls in Lp–metric for $1\le p\le \infty $ . In the last case, extremals are obtained in terms of incomplete Euler integral of the first kind.