# 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

${\displaystyle f(x)<+\infty }$

for at least one x and

${\displaystyle f(x)>-\infty }$

for every x. That is, a convex function is proper if its effective domain is nonempty and it never attains ${\displaystyle -\infty }$.[1] Convex functions that are not proper are called improper convex functions.[2]

A proper concave function is any function g such that ${\displaystyle f=-g}$ 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

${\displaystyle f(x)\geq x\cdot b-\beta }$

for every x.

The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets ${\displaystyle A\subset X}$ and ${\displaystyle B\subset X}$ are non-empty convex sets in the vector space X, then the indicator functions ${\displaystyle I_{A}}$ and ${\displaystyle I_{B}}$ are proper convex functions, but if ${\displaystyle A\cap B=\emptyset }$ then ${\displaystyle I_{A}+I_{B}}$ is identically equal to ${\displaystyle +\infty }$.

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.[3]

## 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.
3. ^ Ioffe, Aleksandr Davidovich; Tikhomirov, Vladimir Mikhaĭlovich (2009), Theory of extremal problems, Studies in Mathematics and its Applications, 6, North-Holland, p. 168, ISBN 9780080875279.