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Concept
Epigraph (mathematics)
Summary
In mathematics, the epigraph or supergraph of a function valued in the extended real numbers is the set, denoted by of all points in the Cartesian product lying on or above its graph. The strict epigraph is the set of points in lying strictly above its graph.
Importantly, although both the graph and epigraph of consists of points in the epigraph consists of points in the subset which is not necessarily true of the graph of
If the function takes as a value then will be a subset of its epigraph
For example, if then the point will belong to but not to
These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa.
The study of continuous real-valued functions in real analysis has traditionally been closely associated with the study of their graphs, which are sets that provide geometric information (and intuition) about these functions. Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions valued in instead of continuous functions valued in a vector space (such as or ). This is because in general, for such functions, geometric intuition is more readily obtained from a function's epigraph than from its graph. Similarly to how graphs are used in real analysis, the epigraph can often be used to give geometrical interpretations of a convex function's properties, to help formulate or prove hypotheses, or to aid in constructing counterexamples.
The definition of the epigraph was inspired by that of the graph of a function, where the of is defined to be the set
The or of a function valued in the extended real numbers is the set
In the union over that appears above on the right hand side of the last line, the set may be interpreted as being a "vertical ray" consisting of and all points in "directly above" it.
Similarly, the set of points on or below the graph of a function is its .
Official source
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