Explicit formulae for geodesics in left invariant sub-Finsler problems on Heisenberg groups via convex trigonometry

TL;DR

This paper derives explicit formulas for geodesics in certain sub-Finsler problems on Heisenberg groups, using convex trigonometry and generalized spherical coordinates, covering various convex unit sets.

Contribution

It introduces a method to explicitly compute geodesics in left-invariant sub-Finsler problems on Heisenberg groups using convex trigonometry and generalized spherical coordinates.

Findings

Explicit geodesic formulas for convex unit sets including $L_p$ balls.

Extremals expressed via incomplete Euler integrals.

Applicable to all left-invariant sub-Riemannian structures on Heisenberg groups.

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 $L_p$-metric for $1\le p\le\infty$. In the last case, extremals are obtained in terms of incomplete Euler integral of the first kind.