Definition 21.2.14A9F1082E9C2

Definition 21.2.1.4A9F1082E9C2 A function $F\colon N_{\mathbb R}\to [-\infty ,+\infty ]$ is convex if its epigraph

is convex. A convex function is proper if it does not take the value $-\infty $ and its effective domain $\operatorname {dom}F=\{ x:F(x)<+\infty \} $ is nonempty; a proper convex function is closed if it is lower semicontinuous. The relative interior $\operatorname {ri}C$ of a convex set $C$ is its interior in its affine span, and $\overline C\setminus \operatorname {ri}C$ is its relative boundary. For a proper convex function $F$ and $x\in \operatorname {dom}F$, the subgradient set of $F$ at $x$ is

Its elements are the subgradients, or supporting slopes, of $F$ at $x$: they are the slopes of the affine functions that lie below $F$ and agree with $F$ at $x$. We set $\partial F(x)=\varnothing $ for $x\notin \operatorname {dom}F$ and $\partial F(E)=\bigcup _{x\in E}\partial F(x)$ for $E\subset N_{\mathbb R}$.