extreme pointExtremePoint2004-06-15 16:43:142006-10-06 01:36:35Definition% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}\begin{defn}
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
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 $z \notin C$.
\end{defn}
For example the set $[0,1] \in {\mathbb{R}}$ is a convex set and $0$ and $1$ are the extreme points.
\begin{thebibliography}{9}
\bibitem{royden}
H.\@ L.\@ Royden. \emph{\PMlinkescapetext{Real Analysis}}. Prentice-Hall, Englewood Cliffs, New Jersey, 1988
\end{thebibliography}