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This characterization of convexity is quite useful to prove the following results. A convex function of one real variable defined on some open interval is continuous on . admits left and right derivatives, and these are monotonically non-decreasing. In addition, the left derivative is left-continuous and the right-derivative is right-continuous.
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.
Convexity is a geometric property with a variety of applications in economics. [1] Informally, an economic phenomenon is convex when "intermediates (or combinations) are better than extremes". For example, an economic agent with convex preferences prefers combinations of goods over having a lot of any one sort of good; this represents a kind of ...
In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.
The classes of s-convex measures form a nested increasing family as s decreases to −∞" . or, equivalently {} {}.Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class.
An example of generalized convexity is orthogonal convexity. [19] A set S in the Euclidean space is called orthogonally convex or ortho-convex, if any segment parallel to any of the coordinate axes connecting two points of S lies totally within S. It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex.
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 optimization problem is defined by two ingredients: [5] [6] The objective function, which is a real-valued convex function of n variables, :;; The feasible set, which is a convex subset.