  This work is an investigation of perimeter measures in the metric measure
space given by the Heisenberg group with Haar measure and a
Carnot-Carath\'eodory metric, which is in general a sub-Finsler metric.
Included is a reduction of Minkowski content in any CC-metric to an integral
formula in terms of Lebesgue surface area in $\mathbb{R}^3$. Using this result,
I study two perimeter measures that arise from the study of Finsler normed
planes, and provide evidence that Pansu's conjecture regarding the
isoperimetric problem in the sub-Riemannian case appears to hold in the more
general sub-Finsler case. This is contrary to the relationship between Finsler
and Riemannian isoperimetrices. In particular, I show that for any CC-metric
there exist a class of surfaces with Legendrian foliation by CC-geodesics
generalizing Pansu's bubble set, but that even in their natural metric using
either perimeter measure, the computed examples of such surfaces have lower
isoperimetric ratio than the Pansu bubble set, which has Legendrian foliation
by sub-Riemannian geodesics.
