Let $x\in\operatorname{stab}(f)$. Then $C_{x}=\{u\in N_{\mathbb{R}}\mid P_{f}(u,x)=0\}.$ In other words, the set $C_{x}$ is characterized by the condition

(3.25)

$$f(u)=\langle x,u\rangle-f^{\vee}(x)\text{ for }u\in C_{x}\quad\text{and}\quad f(u)<\langle x,u\rangle-f^{\vee}(x)\text{ for }u\not\in C_{x}.$$

Thus the restriction of $f$ to $C_{x}$ is an affine function with linear part given by $x$, and $C_{x}$ is the maximal subset where this property holds.