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Exposed point

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The two distinguished points are examples of extreme points of a convex set that are not exposed

In mathematics, an exposed point of a convex set is a point at which some continuous linear functional attains its strict maximum over .[1] Such a functional is then said to expose . There can be many exposing functionals for . The set of exposed points of is usually denoted .

A stronger notion is that of strongly exposed point of which is an exposed point such that some exposing functional of attains its strong maximum over at , i.e. for each sequence we have the following implication: . The set of all strongly exposed points of is usually denoted .

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

References

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  1. ^ Simon, Barry (June 2011). "8. Extreme points and the Krein–Milman theorem" (PDF). Convexity: An Analytic Viewpoint. Cambridge University Press. p. 122. ISBN 9781107007314.