of a function
Let be a function defined on some set with values in the extended real line . Then the supergraph of is the subset of the Cartesian product consisting of all points lying "above" the graph of . The supergraph is sometimes called the epigraph and is denoted by :
The projection of the supergraph onto is known as the effective domain of , denoted by :
The function is said to be proper if
A function defined on a real vector space is convex if and only if is a convex subset of . A function defined on a topological space is upper semi-continuous if and only if is a closed set.
|[a1]||R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) pp. 23; 307|
Supergraph. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Supergraph&oldid=17155