  We study the planar least gradient problem with respect to an anisotropic
norm $\phi$ for continuous boundary data. We prove existence of minimizers for
strictly convex domains $\Omega$. Furthermore, we inspect the issue of
uniqueness and regularity of minimizers only in terms of the modes of convexity
of $\phi$ and $\Omega$. The results are independent from the regularity of
$\phi$.
