View source for Epigraph (mathematics)

{{Short description|Region above a graph}}
{{distinguish|text = epigraph as an inscription studied in the archeological sub-discipline of [[epigraphy]], or [[epigraph (literature)]]}}
[[File:Epigraph.svg|alt=|right|thumb|upright=1.5|Epigraph of a function]]
[[File:Epigraph convex.svg|alt=|right|thumb|upright=1.5|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'''<ref name="NeittaanmäkiRepin2004" /> of a [[Function (mathematics)|function]] <math>f : X \to [-\infty, \infty]</math> valued in the [[Extended real number line|extended real numbers]] <math>[-\infty, \infty] = \Reals \cup \{\pm \infty\}</math> is the [[Set (mathematics)|set]] 
<math display=block>\operatorname{epi} f = \{(x, r) \in X \times \Reals ~:~ r \geq f(x)\}</math>
consisting of all points in the [[Cartesian product]] <math>X \times \Reals</math> lying on or above the function's [[Graph of a function|graph]].{{sfn|Rockafellar|Wets|2009|pp=1-37}} 
Similarly, the '''strict epigraph''' <math>\operatorname{epi}_S f</math> is the set of points in <math>X \times \Reals</math> lying strictly above its graph.

Importantly, unlike the graph of <math>f,</math> the epigraph {{em|always}} consists {{em|entirely}} of points in <math>X \times \Reals</math> (this is true of the graph only when <math>f</math> is real-valued).  
If the function takes <math>\pm \infty</math> as a value then <math>\operatorname{graph} f</math> will {{em|not}} be a subset of its epigraph <math>\operatorname{epi} f.</math> 
For example, if <math>f\left(x_0\right) = \infty</math> then the point <math>\left(x_0, f\left(x_0\right)\right) = \left(x_0, \infty\right)</math> will belong to <math>\operatorname{graph} f</math> but not to <math>\operatorname{epi} f.</math> 
These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa. 

The study of [[Continuous function|continuous]] [[real-valued function]]s in [[real analysis]] has traditionally been closely associated with the study of their [[Graph of a function|graphs]], which are sets that provide geometric information (and intuition) about these functions.{{sfn|Rockafellar|Wets|2009|pp=1-37}} Epigraphs serve this same purpose in the fields of [[convex analysis]] and [[variational analysis]], in which the primary focus is on [[convex function]]s valued in <math>[-\infty, \infty]</math> instead of continuous functions valued in a vector space (such as <math>\Reals</math> or <math>\Reals^2</math>).{{sfn|Rockafellar|Wets|2009|pp=1-37}} This is because in general, for such functions, geometric intuition is more readily obtained from a function's epigraph than from its graph.{{sfn|Rockafellar|Wets|2009|pp=1-37}} 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 [[counterexample]]s.

==Definition==

The definition of the epigraph was inspired by that of the [[graph of a function]], where the {{em|'''{{visible anchor|graph|Graph}}'''}} of <math>f : X \to Y</math> is defined to be the set 
<math display=block>\operatorname{graph} f := \{(x, y) \in X \times Y ~:~ y = f(x)\}.</math>

The {{em|'''{{visible anchor|epigraph|Epigraph}}'''}} or {{em|'''{{visible anchor|supergraph|Supergraph}}'''}} of a function  <math>f : X \to [-\infty, \infty]</math> valued in the [[Extended real number line|extended real numbers]] <math>[-\infty, \infty] = \Reals \cup \{\pm \infty\}</math> is the set{{sfn|Rockafellar|Wets|2009|pp=1-37}}
<math display=block>
\begin{alignat}{4}
\operatorname{epi} f 
&= \{(x, r) \in X \times \Reals ~:~ r \geq f(x)\} \\
&= \left[f^{-1}(- \infty) \times \Reals\right] \cup \bigcup_{x \in f^{-1}(\Reals)} (\{x\} \times [f(x), \infty))
\end{alignat}
</math>
where all sets being unioned in the last line are pairwise disjoint.

In the union over <math>x \in f^{-1}(\Reals)</math> that appears above on the right hand side of the last line, the set <math>\{x\} \times [f(x), \infty)</math> may be interpreted as being a "vertical ray" consisting of <math>(x, f(x))</math> and all points in <math>X \times \Reals</math> "directly above" it. 
Similarly, the set of points on or below the graph of a function is its [[Hypograph (mathematics)|{{visible anchor|hypograph|Hypograph}}]]. 

The {{em|'''{{visible anchor|strict epigraph|Strict epigraph}}'''}} is the epigraph with the graph removed:
<math display=block>
\begin{alignat}{4}
\operatorname{epi}_S f 
&= \{(x, r) \in X \times \Reals ~:~ r > f(x)\} \\
&= \operatorname{epi} f \setminus \operatorname{graph} f \\
&= \bigcup_{x \in X} \left( \{x\} \times (f(x), \infty) \right)
\end{alignat}
</math>
where all sets being unioned in the last line are pairwise disjoint, and some may be empty.

==Relationships with other sets==

Despite the fact that <math>f</math> might take one (or both) of <math>\pm \infty</math> as a value (in which case its graph would {{em|not}} be a subset of <math>X \times \Reals</math>), the epigraph of <math>f</math> is nevertheless defined to be a subset of <math>X \times \Reals</math> rather than of <math>X \times [-\infty, \infty].</math> This is intentional because when <math>X</math> is a [[vector space]] then so is <math>X \times \Reals</math> but <math>X \times [-\infty, \infty]</math> is {{em|never}} a vector space{{sfn|Rockafellar|Wets|2009|pp=1-37}} (since the [[extended real number line]] <math>[-\infty, \infty]</math> is not a vector space). 
This deficiency in <math>X \times [-\infty, \infty]</math> remains even if instead of being a vector space, <math>X</math> 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 of a function|domain]] (rather than the [[codomain]]) of the function is not particularly important for this definition; it can be any [[linear space]]<ref name="NeittaanmäkiRepin2004" /> or even an arbitrary set<ref name="AliprantisBorder2007" /> instead of <math>\Reals^n</math>.

The strict epigraph <math>\operatorname{epi}_S f</math> and the graph <math>\operatorname{graph} f</math> are always disjoint. 

The epigraph of a function <math>f : X \to [-\infty, \infty]</math> is related to its graph and strict epigraph by 
<math display=block>\,\operatorname{epi} f \,\subseteq\, \operatorname{epi}_S f \,\cup\, \operatorname{graph} f</math>
where set equality holds if and only if <math>f</math> is real-valued. However, 
<math display=block>\operatorname{epi} f = \left[\operatorname{epi}_S f \,\cup\, \operatorname{graph} f\right] \,\cap\, [X \times \Reals]</math> 
always holds.

==Reconstructing functions from epigraphs==

The epigraph is [[Empty set|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 <math>f</math> on <math>X</math> be reconstructed from its epigraph <math>E := \operatorname{epi} f</math> (even when <math>f</math> takes on <math>\pm \infty</math> as a value). Given <math>x \in X,</math> the value <math>f(x)</math> can be reconstructed from the intersection <math>E \cap (\{x\} \times \Reals)</math> of <math>E</math> with the "vertical line" <math>\{x\} \times \Reals</math> passing through <math>x</math> as follows: 

<ul>
<li>case 1: <math>E \cap (\{x\} \times \Reals) = \varnothing</math> if and only if <math>f(x) = \infty,</math></li>
<li>case 2: <math>E \cap (\{x\} \times \Reals) = \{x\} \times \Reals</math> if and only if <math>f(x) = -\infty,</math></li>
<li>case 3: otherwise, <math>E \cap (\{x\} \times \Reals)</math> is necessarily of the form <math>\{x\} \times [f(x), \infty),</math> from which the value of <math>f(x)</math> can be obtained by taking the [[infimum]] of the interval.</li>
</ul>

The above observations can be combined to give a single formula for <math>f(x)</math> in terms of <math>E := \operatorname{epi} f.</math> 
Specifically, for any <math>x \in X,</math> 
<math display=block>f(x) = \inf_{} \{r \in \Reals ~:~ (x, r) \in E\}</math>
where by definition, <math>\inf_{} \varnothing := \infty.</math> 
This same formula can also be used to reconstruct <math>f</math> from its strict epigraph <math>E := \operatorname{epi}_S f.</math>

==Relationships between properties of functions and their epigraphs==

A function is [[convex function|convex]] if and only if its epigraph is a [[convex set]]. The epigraph of a real [[affine function]] <math>g : \Reals^n \to \Reals</math> is a [[Half-space (geometry)|halfspace]] in <math>\Reals^{n+1}.</math> 

A function is [[Semi-continuity|lower semicontinuous]] if and only if its epigraph is [[closed set|closed]].

==See also==

* {{annotated link|Effective domain}}
* {{annotated link|Hypograph (mathematics)}}
* {{annotated link|Proper convex function}}

==Citations==
{{commonscat|Epigraph and hypograph (mathematics)|epigraphs und hypographs}}
{{reflist|refs=
<ref name="NeittaanmäkiRepin2004">{{cite book|author1=Pekka Neittaanmäki|author2=Sergey R. Repin|title=Reliable Methods for Computer Simulation: Error Control and Posteriori Estimates|url=https://books.google.com/books?id=s5DA9DerIs4C&pg=PA81|year=2004|publisher=Elsevier|isbn=978-0-08-054050-4|page=81}}</ref>
<ref name="AliprantisBorder2007">{{cite book|author1=Charalambos D. Aliprantis|author2=[[Kim C. Border]]|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|url=https://books.google.com/books?id=4hIq6ExH7NoC&pg=PA8|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-32696-0|page=8|edition=3rd}}</ref>
}}

==References==

* {{Rockafellar Wets Variational Analysis 2009 Springer}} <!--{{sfn|Rockafellar|Wets|2009|p=}}-->
* [[Ralph Tyrell Rockafellar|Rockafellar, Ralph Tyrell]] (1996), ''Convex Analysis'', Princeton University Press, Princeton, NJ. {{ISBN|0-691-01586-4}}.

{{Convex analysis and variational analysis}}

[[Category:Convex analysis]]
[[Category:Mathematical analysis]]
[[Category:Variational analysis]]