Exposed point

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File:Extremenotexposed.pngs of a convex set that are {{em|not}} exposed]]

In mathematics, an exposed point of a convex set $C$ is a point $x\backslash in\; C$ at which some continuous linear functional attains its strict maximum over $C$.{{cite book | last = Simon| first = Barry| authorlink = Barry Simon| title = Convexity: An Analytic Viewpoint| publisher = Cambridge University Press| series = | chapter = 8. Extreme points and the Krein–Milman theorem| chapter-url = http://www.math.caltech.edu/Simon_Chp8.pdf| volume = | edition = | date = June 2011| location = | pages = 122| language = | url = | doi = | id = | isbn = 9781107007314| mr = | zbl = | jfm = }} 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 $\backslash exp(C)$.

A stronger notion is that of strongly exposed point of $C$ which is an exposed point $x\; \backslash in\; C$ such that some exposing functional $f$ of $x$ attains its strong maximum over $C$ at $x$, i.e. for each sequence $(x\_n)\; \backslash subset\; C$ we have the following implication: $f(x\_n)\; \backslash to\; \backslash max\; f(C)\; \backslash Longrightarrow\; \backslash |x\_n\; -x\backslash |\; \backslash to\; 0$. The set of all strongly exposed points of $C$ is usually denoted $\backslash operatorname\{str\}\backslash exp(C)$.

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