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 not necessarily proper or convex. For instance if the sets $A\subset X$ and $B\subset X$ are convex sets in the vector space X, then the indicator functions $I_{A}$ and $I_{B}$ are proper convex functions, but $I_{A}+I_{B}$ is not convex (unless $A\cup B$ is convex), and is possibly identically equal to $+\infty$ if $A\cup B=X$ (i.e. are complementary halfspaces).

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