Let $\|\cdot\|_{2}$ denote the Euclidean norm on $\mathbb{R}^{2}$ and $B_{1}$ the unit ball. Consider the concave function $f\colon B_{1}\to\mathbb{R}$ defined as $f(u)=-\|u\|_{2}$. Then $\operatorname{stab}(f)=\mathbb{R}^{2}$ and the Legendre-Fenchel dual is the function defined by $f^{\vee}(x)=0$ if $\|x\|_{2}\leq 1$ and $f^{\vee}(x)=1-\|x\|_{2}$ otherwise. The decompositions $\Pi(f)$ and $\Pi(f^{\vee})$ consist of a collection of pieces of three different types and the Legendre-Fenchel correspondence ${\mathcal{L}}f\colon\Pi(f)\to\Pi(f^{\vee})$ is given, for $z\in S^{1}$, by