from Theorem \[twierdzeniezbgg\] $E$ is a minimal set and only finitely many connected components of the boundary of a minimal set may intersect any compact subset of $\Omega$.

In the second case we may without loss of generality assume that $\overline{xy}$ and $\overline{xz}$ are adjacent. Consider the function $\chi_{E_t}$. In the area enclosed by the intervals $\overline{xy}, \overline{xz}$ and the arc $\overline{yz} \subset \partial\Omega$ not containing $x$ we have $\chi_{E_t} = 1$ and $\chi_{E_t} = 0$ on the two sides of the triangle (or the opposite situation, which we handle similarly). Then $\chi_{E_t}$ is not a function of least gradient: the function $\widetilde{\chi_{E_t}} = \chi_{E_t} - \chi_{\Delta xyz}$ has strictly smaller total variation due to the triangle inequality. This contradicts Theorem \[twierdzeniezbgg\].

The result above is sharp. As the following example shows, we may not relax the assumption of convexity of $\Omega$.

Denote by $\varphi$ the angular coordinate in the polar coordinates on the plane. Let $\Omega = B(0,1) \backslash (\{ \frac{\pi}{4} \leq \varphi \leq \frac{3\pi}{4} \} \cup \{ 0 \}) \subset \mathbb{R}^2$, i.e. the unit ball with one quarter removed. Take the boundary data $f \in L^1(\partial \Omega)$ to be $$f(x,y) = \twopartdef{1}{y \geq 0}{0}{y < 0.}$$ Then the 