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then is called strictly convex. [1]Convex functions are related to convex sets. Specifically, the function is convex if and only if its epigraph. A function (in black) is convex if and only if its epigraph, which is the region above its graph (in green), is a convex set.
This is a generalization of the concept of strongly convex function; by taking () = we recover the definition of strong convexity. It is worth noting that some authors require the modulus ϕ {\displaystyle \phi } to be an increasing function, [ 17 ] but this condition is not required by all authors.
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 +.
A plane curve is the image of any continuous function from an interval to the Euclidean plane.Intuitively, it is a set of points that could be traced out by a moving point. More specifically, smooth curves generally at least require that the function from the interval to the plane be continuously differentiable, and in some contexts are defined to require higher derivative
In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces ¯, (,) in quantum cohomology. [ 1 ] : §1 [ 2 ] [ 3 ] These moduli spaces are smooth orbifolds whenever the target space is 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.
In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions.
If : is a continuous function and is closed, then is closed.; 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 .