Epigraph (mathematics)
In mathematics, the epigraph or supergraph[1] of a function
valued in the extended real numbers
consisting of all points in the Cartesian product
[2] Similarly, the strict epigraph
lying strictly above its graph.
Importantly, unlike the graph of
the epigraph always consists entirely of points in
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
[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.
valued in the extended real numbers
where all sets being unioned in the last line are pairwise disjoint.
that appears above on the right hand side of the last line, the set
may be interpreted as being a "vertical ray" consisting of
Similarly, the set of points on or below the graph of a function is its hypograph.
where all sets being unioned in the last line are pairwise disjoint, and some may be empty.
as a value (in which case its graph would not be a subset of
is never a vector space[2] (since the extended real number line
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
is related to its graph and strict epigraph by
The epigraph is empty if and only if the function is identically equal to infinity.
as follows: The above observations can be combined to give a single formula for
The epigraph of a real affine function
A function is lower semicontinuous if and only if its epigraph is closed.
