Effective domain

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, ${\displaystyle f:X\to \mathbb {R} \cup \{\pm \infty \}}$, 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 ${\displaystyle f:X\to \mathbb {R} \cup \{\pm \infty \}}$ onto X. That is

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