open, bounded, strictly convex set with Lipschitz boundary. Suppose $u \in BV(\Omega)$ is a function of least gradient. Then there exist functions $u_c, u_j \in BV(\Omega)$ such that $u = u_c + u_j$ and $(Du)_c = Du_c$ and $(Du)_j = Du_j$, i.e. one can represent $u$ as a sum of a continuous function and a piecewise constant function. They are of least gradient in $\Omega$. Moreover this decomposition is unique up to an additive constant.

The final chapter takes on the subject of anisotropy. As it was proved in [@JMN], for an anisotropic norm $\phi$ on $\mathbb{R}^N$ smooth with respect to the Euclidean norm there is a unique solution to the anisotropic LGP. I consider $p-$norms on the plane for $p \in [1, \infty]$ to show that for $p = 1, \infty$, i.e. where the anisotropy is not smooth, the solutions need not be unique even for smooth boundary data (see Examples \[ex:l1\] and \[ex:linfty\]), whereas for $1 < p < \infty$, when the anisotropy is smooth, Theorem \[tw:anizotropia\] asserts that the only connected minimal surface with respect to the p-norm is a line segment, similarly to the isotropic solution.

Let $\Omega \subset \mathbb{R}^2$ be an open convex set. Let 