The recession function of a concave function $f\colon N_{\mathbb{R}}\to{\underline{\mathbb{R}}}$, denoted $\operatorname{rec}(f)$, is the function

$$\operatorname{rec}(f)\colon N_{\mathbb{R}}\longrightarrow{\underline{\mathbb{R}}},\quad u\longmapsto\inf_{v\in{\operatorname{dom}}(f)}(f(u+v)-f(v)).$$

This is a concave conical function. If $f$ is closed, its recession function can be defined as the limit

(3.49)

$$\operatorname{rec}(f)(u)=\lim_{\lambda\to\infty}\lambda^{-1}f(v_{0}+\lambda u)$$

for any $v_{0}\in{\operatorname{dom}}(f)$ [Roc70, Theorem 8.5].