Definition 21.11.34FA12A948A00

Definition 21.11.3.4FA12A948A00 A convex function on $N$ is a function $F\colon N\rightarrow [-\infty ,\infty ]$ such that the epigraph $\operatorname{epi}F$ is a convex subset of $N\times \mathbb {R}$.

The effective domain of $F$ is the set

A convex function $F$ on $N$ such that $\operatorname{Dom}F\neq \varnothing $ and $F(n)\neq -\infty $ for all $n\in N$ is said to be proper.

The set of convex functions on $N$ is denoted by $\mathrm{Conv}(N)$. The subset of proper convex functions is denoted by $\mathrm{Conv}^{\mathrm{prop}}(N)$.