A geometric proof of regularity of all anisotropic minimal surfaces in ${\mathbb{R}}^2$

A set of locally finite perimeter $E\subset {\mathbb{R}}^n$ is called an anisotropic minimal surface in an open set $A$ if $\mathrm{\Phi }(E;A)\le \mathrm{\Phi }(F;A)$ for some surface energy $\Phi(E;A) = \int_{\partial^{*}E \cap A} | \nu_{E}| d $\mathcal{H}$^{n-1}$andallsetsoflocallyfiniteperimeter$F$suchthat$E \Delta F \subset \subset A$. In this short note we provide the details of a geometric proof verifying that all anisotropic surface minimizers in ${\mathbb{R}}^2$ whose corresponding integrand $\mathrm{\parallel }\cdot \mathrm{\parallel }$ is strictly convex are locally disjoint unions of line segments. This demonstrates that, in the plane, strict convexity of $| \cdot |$ is both necessary and sufficient for regularity. The corresponding Bernstein theorem is also proven: global anisotropic minimizers $E \subset $\mathbb{R}$^{2}$ are half-spaces.