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

$f(x) \ge x \cdot b - \beta$

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 non-empty convex sets in the vector space X, then the indicator 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.