Closed convex function

A closed convex function is an extended-real-valued function $f:\mathbb{R}^n\to(-\infty,+\infty]$ that is convex and has a closed epigraph

so that $\mathrm{epi}(f)$ is a closed convex set in $\mathbb{R}^n\times\mathbb{R}$.

Equivalently, $f$ is closed convex iff it is convex and lower semicontinuous, meaning $\liminf_{x\to x_0} f(x)\ge f(x_0)$ for every $x_0$. Closedness is the regularity condition that makes conjugation behave well: for proper functions, closed convexity is exactly the condition for equality $f=f^{**}$ in the biconjugate construction (see Fenchel–Moreau theorem ).

Examples:

• The norm $f(x)=\|x\|_2$ is closed and convex on $\mathbb{R}^n$.
• If $C$ is a closed convex set, then the indicator $\delta_C$ is a closed convex function; if $C$ is convex but not closed (e.g. an open ball), then $\delta_C$ is convex but not closed.