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The extended-real function that is zero on a set and positive infinity outside it.
Let be a set and let . The indicator function of is the extended-real-valued function defined by
Its effective domain is , and its epigraph is .
Indicator functions encode constraints as penalties: minimizing is equivalent to minimizing subject to .
When is a real vector space, is convex if and only if is a convex set.
The extended real number system adjoins two formal endpoints to the real numbers:
ordered by for every . Convex analysis often uses the one-sided extension
for functions that encode infeasible points by the value .
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 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 (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.