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Hypograph of a function. In mathematics, the hypograph or subgraph of a function: is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph.
Epigraph of a function A function (in black) is convex if and only if the region above its graph (in green) is a convex set.This region is the function's epigraph. In mathematics, the epigraph or supergraph [1] of a function: [,] valued in the extended real numbers [,] = {} is the set = {(,) : ()} consisting of all points in the Cartesian product lying on or above the function's graph. [2]
Epigraph (mathematics) - for a function f : Rn→R, [check spelling] the set of points lying on or above its graph; Extreme point - for a convex set S in a real vector space, a point in S which does not lie in any open line segment joining two points of S. Fenchel conjugate; Fenchel's inequality
The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.
Equivalently, if the epigraph defined by = {(,) + |, ()} is closed, then the function is closed. This definition is valid for any function, but most used for convex functions . A proper convex function is closed if and only if it is lower semi-continuous .
What is typically used is y vs. x, such that x is horizontal and y is vertical. However, when specifically talking about plotting a function vs. its input, it is more clear and intuitive to plot f(x) vs. x (or f(y) vs. y or whatever), since the variables x and y are just placeholders. EmergencyBackupChicken 17:00, 7 May 2009 (UTC)
A coin without an epigraph or inscription. Many ancient coins used only a simple picture of an animal to show value or weight. annealing The process of repeatedly heating and cooling metal in order to relieve stresses. This is often done with coin blanks to make the metal less brittle before striking. assay
Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.