Proper convex function


In mathematical analysis and optimization, a proper convex function is a convex function f taking values in the extended real number line such that
for at least one x and
for every x. That is, a convex function is proper if its effective domain is nonempty and it never attains. Convex functions that are not proper are called improper convex functions.
A proper concave function is any function g such that 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
for every x.
The sum of two proper convex functions is convex, but not necessarily proper. For instance if the sets and are non-empty convex sets in the vector space X, then the characteristic functions and are proper convex functions, but if then is identically equal to.
The infimal convolution of two proper convex functions is convex but not necessarily proper convex.