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Epigraph (mathematics)

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] Similarly, the strict epigraph is the set of points in lying strictly above its 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.

Importantly, unlike the graph of the epigraph always consists entirely of points in (this is true of the graph only when is real-valued). If the function takes as a value then will not 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.[2] 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 ).[2] This is because in general, for such functions, geometric intuition is more readily obtained from a function's epigraph than from its graph.[2] 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.

Definition

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The definition of the epigraph was inspired by that of the graph of a function, where the graph of   is defined to be the set  

The epigraph or supergraph of a function   valued in the extended real numbers   is the set[2]   where all sets being unioned in the last line are pairwise disjoint.

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 hypograph.

The strict epigraph is the epigraph with the graph removed:   where all sets being unioned in the last line are pairwise disjoint, and some may be empty.

Relationships with other sets

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Despite the fact that   might take one (or both) of   as a value (in which case its graph would not be a subset of  ), the epigraph of   is nevertheless defined to be a subset of   rather than of   This is intentional because when   is a vector space then so is   but   is never a vector space[2] (since the extended real number line   is not a vector space). This deficiency in   remains even if instead of being a vector space,   is merely a non-empty subset of some vector space. The epigraph being a subset of a vector space allows for tools related to real analysis and functional analysis (and other fields) to be more readily applied.

The domain (rather than the codomain) of the function is not particularly important for this definition; it can be any linear space[1] or even an arbitrary set[3] instead of  .

The strict epigraph   and the graph   are always disjoint.

The epigraph of a function   is related to its graph and strict epigraph by   where set equality holds if and only if   is real-valued. However,   always holds.

Reconstructing functions from epigraphs

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The epigraph is empty if and only if the function is identically equal to infinity.

Just as any function can be reconstructed from its graph, so too can any extended real-valued function   on   be reconstructed from its epigraph   (even when   takes on   as a value). Given   the value   can be reconstructed from the intersection   of   with the "vertical line"   passing through   as follows:

  • case 1:   if and only if  
  • case 2:   if and only if  
  • case 3: otherwise,   is necessarily of the form   from which the value of   can be obtained by taking the infimum of the interval.

The above observations can be combined to give a single formula for   in terms of   Specifically, for any     where by definition,   This same formula can also be used to reconstruct   from its strict epigraph  

Relationships between properties of functions and their epigraphs

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A function is convex if and only if its epigraph is a convex set. The epigraph of a real affine function   is a halfspace in  

A function is lower semicontinuous if and only if its epigraph is closed.

See also

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Citations

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  1. ^ a b Pekka Neittaanmäki; Sergey R. Repin (2004). Reliable Methods for Computer Simulation: Error Control and Posteriori Estimates. Elsevier. p. 81. ISBN 978-0-08-054050-4.
  2. ^ a b c d e f Rockafellar & Wets 2009, pp. 1–37.
  3. ^ Charalambos D. Aliprantis; Kim C. Border (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer Science & Business Media. p. 8. ISBN 978-3-540-32696-0.

References

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