\partial \{ u > t \}$, contradiction. Similarly suppose two such intervals intersect in $x \in \Omega$. Then the measure-theoretic normal vector at $x$ has length smaller then $1$, depending on the angle between the two intervals. Thus $x \notin \partial^* \{ u > t \}$, contradiction.

If we additionally assume that $\Omega$ is convex, then the union is disjoint also on $\partial\Omega$:

\[slabazasadamaksimum2\] Let $\Omega \subset \mathbb{R}^2$ be an open, bounded, convex set with Lipschitz boundary 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 intervals, pairwise disjoint in $\overline{\Omega}$, such that every interval connects two points of $\partial \Omega$.

Suppose that at least two intervals in $\partial E_t$ end in $x \in \partial\Omega$: $\overline{xy}$ and $\overline{xz}$. We have two possibilities: there are countably many intervals in $\partial E_t$, which end in $x$, with the other end lying in the arc $\overline{yz} \subset \partial\Omega$ which does not contain $x$; or there are finitely many. The first case is excluded by the monotonicity formula for minimal surfaces, see for example [@Sim Theorem 17.6, Remark 37.9], as 