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A function (in black) is convex if and only if the region above its graph (in green) is a convex set. A graph of the bivariate convex function x 2 + xy + y 2. Convex vs. Not convex
If : is a continuous function and is open, then is closed if and only if it converges to along every sequence converging to a boundary point of . [ 2 ] A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that h ≤ f (called the affine minorants of f ).
For every proper convex function : [,], there exist some and such that ()for every .. The sum of two proper convex functions is convex, but not necessarily proper. [4] For instance if the sets and are non-empty convex sets in the vector space, then the characteristic functions and are proper convex functions, but if = then + is identically equal to +.
Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function and the function , which is by definition convex.However, being logarithmically convex is a strictly stronger property than being convex.
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
Strictly convex function, a function having the line between any two points above its graph; Strictly convex polygon, a polygon enclosing a strictly convex set of points; Strictly convex set, a set whose interior contains the line between any two points; Strictly convex space, a normed vector space for which the closed unit ball is a strictly ...