In mathematics, an exposed point of a

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

C

is a point

x\in C

at which some

continuous linear functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear ...

attains its strict

maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...

over

C

. Such a functional is then said to ''expose''

x

. There can be many exposing functionals for

x

. The set of exposed points of

C

is usually denoted

\exp(C)

. A stronger notion is that of ''strongly exposed point'' of

C

which is an exposed point

x \in C

such that some exposing functional

f

x

attains its strong maximum over

C

x

, i.e. for each sequence

(x_n) \subset C

we have the following implication:

f(x_n) \to \max f(C) \Longrightarrow \, x_n -x\,  \to 0

. The set of all strongly exposed points of

C

is usually denoted

\operatorname\exp(C)

. There are two weaker notions, that of

extreme point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or ...

and that of support point of

C

References

Mathematical analysis Convex geometry Functional analysis {{mathanalysis-stub