Epigraph (mathematics)
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. Definition

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 . The  is the epigraph with the graph removed:

Relationships with other sets

Despite the fact that  might take one (or both) of  as a value (in which case its graph would  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  a vector space (since the extended real number line  is not a vector space). More generally, if  is only a non-empty subset of some vector space then  is never even a  of  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 or even an arbitrary set 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

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

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

Citations

References

  
 Rockafellar, Ralph Tyrell (1996), Convex Analysis, Princeton University Press, Princeton, NJ. . Convex analysis
Mathematical analysis
Variational analysis