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

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 .[1] Convex functions that are not proper are called improper convex functions.[2]

A proper concave function is any function g such that is a proper convex function.


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.[3] 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.[4]


  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. Boyd, Stephen (2004). Convex Optimization. Cambridge, UK: Cambridge University Press. p. 79. ISBN 978-0-521-83378-3.
  4. Ioffe, Aleksandr Davidovich; Tikhomirov, Vladimir Mikhaĭlovich (2009), Theory of extremal problems, Studies in Mathematics and its Applications, 6, North-Holland, p. 168, ISBN 9780080875279.
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