| Version current |
Version 3 |
| \begin{defn} |
\begin{defn} |
| Let $C$ be a convex subset of a vector space $X$. A point $x \in C$ is |
Let $C$ be a convex subset of a vector space $X$. A point $x \in C$ is |
| called an {\em extreme point} if it is not an interior point of any line segment |
called an {\em extreme point} if it is not an interior point of any line segment |
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in $C$. That is $x$ is extreme if and only if whenever $x = ty +(1-t)z$, $t \in (0,1)$, $z \not= y$, implies either $y \notin C$ or $z \notin C$.
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in $C$. That is $x$ is extreme if and only if whenever $x = ty +(1-t)z$, $t \in (0,1)$ implies either $y \notin C$ or $z \notin C$.
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| \end{defn} |
\end{defn} |
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| For example the set $[0,1] \in {\mathbb{R}}$ is a convex set and $0$ and $1$ are the extreme points. |
For example the set $[0,1] \in {\mathbb{R}}$ is a convex set and $0$ and $1$ are the extreme points. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{royden} |
\bibitem{royden} |
| H.\@ L.\@ Royden. \emph{\PMlinkescapetext{Real Analysis}}. Prentice-Hall, Englewood Cliffs, New Jersey, 1988 |
H.\@ L.\@ Royden. \emph{\PMlinkescapetext{Real Analysis}}. Prentice-Hall, Englewood Cliffs, New Jersey, 1988 |
| \end{thebibliography} |
\end{thebibliography} |
