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