solution to the least gradient problem is the function (defined inside $\Omega$) $$u(x,y) = \twopartdef{1}{y \geq 0}{0}{y < 0,}$$ in particular $\partial \{ u \geq 1 \}$ consists of two horizontal line segments whose closures intersect at the point $(0,0) \in \partial\Omega$. Note that in this example the set $\Omega$ is star-shaped, but it is not convex.

In higher dimensions, we are going to need a result from [@SWZ] concerning minimal surfaces:

\[stw:sternbergpowmin\] $($[@SWZ Theorem 2.2])\
Suppose $E_1 \subset E_2$ and let $\partial E_1, \partial E_2$ are area-minimizing in a open set $U$. Further, suppose $x \in \partial E_1 \cap \partial E_2 \cap U$. Then $\partial E_1$ and $\partial E_2$ agree in some neighbourhood of $x$.

$($weak maximum principle$)$\
Let $\Omega \subset \mathbb{R}^N$, where $N \leq 7$ and suppose $u \in BV(\Omega)$ is a function of least gradient. Then for every $t \in \mathbb{R}$ the set $\partial \{ u > t \}$ is empty or it is a sum of minimal surfaces $S_{t,i}$, pairwise disjoint in $\Omega$, which satisfy $\partial S_{t,i} \subset \partial \Omega$.

Let us notice, that with only subtle changes the previous proof works also in the case $N \leq 7$, i.e. when boundaries of superlevel sets are 