# extreme point

###### Definition.

Let $C$ be a convex subset of a vector space $X$. A point $x\in C$ is called an 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 $z\notin C$.

For example the set $[0,1]\in{\mathbb{R}}$ is a convex set and $0$ and $1$ are the extreme points.

## References

• 1 H. L. Royden. . Prentice-Hall, Englewood Cliffs, New Jersey, 1988
Title extreme point ExtremePoint 2013-03-22 14:24:55 2013-03-22 14:24:55 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 52A99 FaceOfAConvexSet ExposedPointsAreDenseInTheExtremePoints