In the Heisenberg group $\mathbb{H}^n$ we consider a notion of perimeter
associated to a convex body $K\subset \mathbb{R}^{2n}$ of class $C^2_+$
containing $0$ in its interior. By means of the first variation formula,
we define a natural notion of sub-Finsler mean curvature of a $t$-graph
and study the sub-Finsler prescribed mean curvature equation on a
bounded domain $\Omega$ in the hyperplane $\{t=0\}$ that results when
imposing that the vertical graph of a function over $\Omega$ is a
hypersurface with prescribed mean curvature $H$. When $H$ is constant
and strictly smaller than the anisotropic mean curvature of $\partial
\Omega$, we prove the existence of a Lipschitz solution to the Dirichlet
problem for the sub-Finsler CMC equation by means of an approximation
scheme. These results were obtained in a joint work with Giovannardi,
Pinamonti and Verzellesi.
