  We consider an asymmetric left-invariant norm $||\cdot ||_K$ in the first
Heisenberg group $\mathbb{H}^1$ induced by a convex body $K\subset\mathbb{R}^2$
containing the origin in its interior. Associated to $\|\cdot\|_K$ there is a
perimeter functional, that coincides with the classical sub-Riemannian
perimeter in case $K$ is the closed unit disk centered at the origin of
$\mathbb{R}^2$. Under the assumption that $K$ has $C^2$ boundary with strictly
positive geodesic curvature we compute the first variation formula of perimeter
for sets with $C^2$ boundary. The localization of the variational formula in
the non-singular part of the boundary, composed of the points where the tangent
plane is not horizontal, allows us to define a mean curvature function $H_K$
out of the singular set. In the case of non-vanishing mean curvature, the
condition that $H_K$ be constant implies that the non-singular portion of the
boundary is foliated by horizontal liftings of translations of $\partial K$
dilated by a factor of $1/H_K$. Based on this we can defined a sphere
$\mathbb{S}_K$ with constant mean curvature $1$ by considering the union of all
horizontal liftings of $\partial K$ starting from $(0,0,0)$ until they meet
again in a point of the vertical axis. We give some geometric properties of
this sphere and, moreover, we prove that, up to non-homogeneous dilations and
left-translations, they are the only solutions of the sub-Finsler isoperimetric
problem in a restricted class of sets.
