Effective domain

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In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function.

Given a vector space X then a convex function mapping to the extended reals, , has an effective domain defined by


If the function is concave, then the effective domain is


The effective domain is equivalent to the projection of the epigraph of a function onto X. That is


Note that if a convex function is mapping to the normal real number line given by then the effective domain is the same as the normal definition of the domain.

A function is a proper convex function if and only if f is convex, the effective domain of f is nonempty and for every .[3]


  1. ^ a b 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. ^ Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. p. 400. ISBN 978-3-11-018346-7. 
  3. ^ a b Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 23. ISBN 978-0-691-01586-6.