Infinite Geodesics of Sub-Finsler Distances in Heisenberg Groups

We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [8] and has applications in the characterization of isometric embeddings into Heisenberg groups.