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The effective domain of an extended-real-valued convex function is a convex set.
Let be a real vector space and let be a convex function. Then its effective domain
is a convex set.
If and , convexity gives
Thus .
A vector space over a field is a set with operations (functions) and , and an element , satisfying the following for all and :
Let be a real vector space and let be an extended-real-valued function.
The function is convex if its epigraph
is a convex set in .
Let be a vector space and let be an extended-real-valued function.
The function is proper if . With the stated codomain, this is equivalent to the usual requirement that is never and is not identically .
Let be a vector space (typically over ). A set is convex if for all and all we have
Context. Equivalently, is convex iff it contains every line segment joining any two of its points. Convexity is the core geometric notion underlying convex analysis and optimization.