Exposed point explained

C is a point x\in C at which some continuous linear functional attains its strict maximum over C .^[1] 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 of x attains its strong maximum over C at 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{str}\exp(C) .

There are two weaker notions, that of extreme point and that of support point of

C .

Notes and References

1. Book: Simon, Barry. Barry Simon

   . Barry Simon. Convexity: An Analytic Viewpoint. Cambridge University Press. 8. Extreme points and the Krein–Milman theorem. http://www.math.caltech.edu/Simon_Chp8.pdf. June 2011. 122. 9781107007314.