Geodesics in the Heisenberg group
  We provide a new and elementary proof for the structure of geodesics in the Heisenberg group $\mathbb{H}^n$. The proof is based on a new isoperimetric inequality for closed curves in $\mathbb{R}^{2n}$. We also prove that the Carnot-Carath\'eodory metric is real analytic away from the center of the group. 
Piotr Haj{\l}asz and Scott Zimmerman
math.MG