Supergraph
of a function
Let $ f: X \rightarrow \overline{\mathbf R}\; $ be a function defined on some set $ X $ with values in the extended real line $ \overline{\mathbf R}\; = \mathbf R \cup \{ - \infty \} \cup \{ + \infty \} $. Then the supergraph of $ f $ is the subset of the Cartesian product $ X \times \mathbf R $ consisting of all points $ ( x, \alpha ) $ lying "above" the graph of $ f $. The supergraph is sometimes called the epigraph and is denoted by $ \mathop{\rm epi} f $:
$$
\mathop{\rm epi} f = \
\{ {( x, \alpha ) \in X \times \mathbf R } : {\alpha \geq f ( x) } \} . $$
The projection of the supergraph onto $ X $ is known as the effective domain of $ f $, denoted by $ \mathop{\rm dom} f $:
$$
\mathop{\rm dom} f = \{ {x \in X } : {f ( x) < + \infty } \}
. $$
The function $ f $ is said to be proper if
$$ f ( x) > - \infty , \forall x,\ \textrm{ and } \ \
\mathop{\rm dom} f \neq \emptyset .
$$
A function $ f: X \rightarrow \mathbf R $ defined on a real vector space $ X $ is convex if and only if $ \mathop{\rm epi} f $ is a convex subset of $ X \times \mathbf R $. A function $ f: X \rightarrow \overline{\mathbf R}\; $ defined on a topological space $ X $ is upper semi-continuous if and only if $ \mathop{\rm epi} f $ is a closed set.
Comments
References
| [a1] | R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) pp. 23; 307 |
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