# 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 $-\infty$.[1] Convex functions that are not proper are called improper convex functions.[2]

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 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 identically equal to $+\infty$ if $A \cap B = \emptyset$.

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.[citation needed]

## References

1. ^ Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 254. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
2. ^ Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 24. ISBN 978-0-691-01586-6.