Epigraph (mathematics)

In mathematics, the epigraph or supergraph of a function

f

:

X

→

[

−

∞

,

∞

]

{\displaystyle f:X\to [-\infty ,\infty ]}

valued in the extended real numbers

[

−

∞

,

∞

]

=

R

∪

{

±

∞

}

{\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}

is the set

epi

⁡

f

=

{

(

x

,

r

)

∈

X

×

R

:

r

≥

f

(

x

)

}

{\displaystyle \operatorname {epi} f=\{(x,r)\in X\times \mathbb {R} ~:~r\geq f(x)\}}

consisting of all points in the Cartesian product

X

×

R

{\displaystyle X\times \mathbb {R} }

lying on or above the function's graph.

Similarly, the strict epigraph

epi

S

⁡

f

{\displaystyle \operatorname {epi} _{S}f}

is the set of points in

X

×

R

{\displaystyle X\times \mathbb {R} }

lying strictly above its graph.

Importantly, unlike the graph of

f

,

{\displaystyle f,}

the epigraph always consists entirely of points in

X

×

R

{\displaystyle X\times \mathbb {R} }

(this is true of the graph only when

f

{\displaystyle f}

is real-valued).

If the function takes

±

∞

{\displaystyle \pm \infty }

as a value then

graph

⁡

f

{\displaystyle \operatorname {graph} f}

will not be a subset of its epigraph

epi

⁡

f

.

{\displaystyle \operatorname {epi} f.}

For example, if

f

(

x

0

)

=

∞

{\displaystyle f\left(x_{0}\right)=\infty }

then the point

(

x

0

,

f

(

x

0

)

)

=

(

x

0

,

∞

)

{\displaystyle \left(x_{0},f\left(x_{0}\right)\right)=\left(x_{0},\infty \right)}

will belong to

graph

⁡

f

{\displaystyle \operatorname {graph} f}

but not to

epi

⁡

f

.

{\displaystyle \operatorname {epi} f.}

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

[

−

∞

,

∞

]

{\displaystyle [-\infty ,\infty ]}

instead of continuous functions valued in a vector space (such as

R

{\displaystyle \mathbb {R} }

or

R

2

{\displaystyle \mathbb {R} ^{2}}

). 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.