Proper, convex, and lower-semicontinuous extended-real functions on Euclidean space

Fix an integer $d\ge 1$. Write $\bar{\mathbb{R}}=\mathbb{R}\cup \{+\mathrm{\infty }\}$ with its usual order and extended addition; in a convex combination set $0\cdot (+\mathrm{\infty })=0$ and $\lambda \cdot (+\mathrm{\infty })=+\mathrm{\infty }$ for $\lambda >0$. A function $f:{\mathbb{R}}^d\to \bar{\mathbb{R}}$ is proper when it never takes the value $-\mathrm{\infty }$ and its effective domain

$\mathrm{dom}f:=\{x\in {\mathbb{R}}^d:f(x)<+\mathrm{\infty }\}$

is nonempty. It is convex when

$f((1-t)x+ty)\le (1-t)f(x)+tf(y)(x,y\in {\mathbb{R}}^d,t\in [0,1])\mathrm{.}$

It is lower semicontinuous when each weak sublevel set $\{x:f(x)\le a\}$ is closed in the Euclidean topology, for every $a\in \mathbb{R}$. Its epigraph is

$\mathrm{epi}f:=\{(x,r)\in {\mathbb{R}}^d\times \mathbb{R}:f(x)\le r\}\mathrm{.}$

On this page “closed proper convex” means proper, convex, and lower semicontinuous in these conventions. The pairing of $p,x\in {\mathbb{R}}^d$ is the Euclidean inner product $⟨p,x⟩$ of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.