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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]
Consider a real number, and the relation .Then the upper contour set of would be the set of numbers that were greater than or equal to ,; the strict upper contour set of would be the set of numbers that were greater than ,
A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.. Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces).
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