# Proper convex function

In mathematical analysis (in particular convex analysis) and optimization, a proper convex function is a convex function f taking values in the extended real number line such that

$f(x)<+\infty$ for at least one x and

$f(x)>-\infty$ for every x. That is, a convex function is proper if its effective domain is nonempty and it never attains $-\infty$ . Convex functions that are not proper are called improper convex functions.

A proper concave function is any function g such that $f=-g$ is a proper convex function.

## Properties

For every proper convex function f on Rn there exist some b in Rn and β in R such that

$f(x)\geq x\cdot b-\beta$ for every x.

The sum of two proper convex functions is convex, but not necessarily proper. For instance if the sets $A\subset X$ and $B\subset X$ are non-empty convex sets in the vector space X, then the characteristic functions $I_{A}$ and $I_{B}$ are proper convex functions, but if $A\cap B=\emptyset$ then $I_{A}+I_{B}$ is identically equal to $+\infty$ .

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.