Find a concept
Start typing to search the mathematical index.
The effective domain dom(f) of a convex function is a convex set
Corollary. If is a convex function on a vector space , then its domain is a convex set.
Connection. This follows immediately from Jensen's inequality in Theorem (equivalent characterizations): if , then the right-hand side is finite, forcing .
Let be a vector space and let be an extended-real-valued function.
The function is convex if its epigraph (see epigraph) is a convex set in .
Context. This geometric definition is equivalent to analytic inequalities such as Jensen's inequality; see equivalent characterizations of convexity.
Let be a vector space and let be an extended-real-valued function (see extended reals).
The function is proper if .
Context. The epigraph turns function properties into geometric properties of sets; convexity of is defined by convexity of (see convex functions via epigraphs).
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.
Theorem. Let be an extended-real-valued function on a vector space . The following are equivalent:
Context. Item (3) says that convexity is equivalent to "subadditivity under" convex combinations of finitely many points.
Proof sketch. (2) ⇒ (1): show any convex combination of points in the epigraph stays in the epigraph using the inequality. (1) ⇒ (2): apply convexity of the epigraph to and . (2) ⇔ (3): (3) implies (2) by taking ; (2) implies (3) by induction and the convex-combination characterization.