Epigraph (mathematics)
In mathematics, the epigraph or supergraph of a function formula_1 valued in the extended real numbers formula_2 is the set, denoted by formula_3 of all points in the Cartesian product formula_4 lying on or above its graph. The strict epigraph formula_5 is the set of points in formula_4 lying strictly above its graph. Importantly, although both the graph and epigraph of formula_7 consists of points in formula_8 the epigraph consists of points in the subset formula_9 which is not necessarily true of the graph of formula_10 If the function takes formula_11 as a value then formula_12 will be  a subset of its epigraph formula_13 For example, if formula_14 then the point formula_15 will belong to formula_12 but not to formula_13 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  formula_18 instead of continuous functions valued in a vector space (such as formula_19 or formula_20). 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 formula_21 is defined to be the  set The or of a function formula_1 valued in the extended real numbers formula_2 is the set In the union over formula_26 that appears above on the right hand side of the last line, the set formula_27 may be interpreted as being a "vertical ray" consisting of formula_28 and all points in formula_4 "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 formula_7 might take one (or both) of formula_11 as a value (in which  case its graph would be a subset of formula_4), the epigraph of formula_7 is nevertheless defined to be a subset of formula_4 rather than of formula_36 This is intentional because when formula_37 is a vector space then so is formula_4 but formula_39 is a vector space (since the extended real number line formula_18 is not a vector space). More generally, if formula_37 is only a non-empty subset of some vector space then formula_39 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 formula_43. The strict epigraph formula_5 and the graph formula_12 are always disjoint. The epigraph of a function formula_1 is related to its graph and strict epigraph by where set equality holds if and only if formula_7 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 formula_7 on formula_37 be reconstructed from its epigraph formula_52 (even when formula_7 takes on formula_11 as a value). Given formula_55 the value formula_56 can be reconstructed from the intersection formula_57 of formula_58 with the "vertical line" formula_59 passing through formula_60 as follows: The above observations can be combined to give a single formula for formula_56 in terms of formula_62 Specifically, for any formula_55 where by definition, formula_65 This same formula can also be used to reconstruct formula_7 from its strict epigraph formula_67 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 formula_68 is a halfspace in formula_69 A function is lower semicontinuous if and only if its epigraph is closed.