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extreme point (Definition)
Definition 1   Let $C$ be a convex subset of a vector space $X$ . A point $x \in C$ is called an 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 $z \notin C$ .

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

Bibliography

1
H. L. Royden. Real Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1988




"extreme point" is owned by jirka.
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See Also: face of a convex set, exposed points are dense in the extreme points

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Cross-references: convex set, implies, line segment, interior point, point, vector space, convex subset
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This is version 4 of extreme point, born on 2004-06-15, modified 2006-10-06.
Object id is 5920, canonical name is ExtremePoint.
Accessed 6235 times total.

Classification:
AMS MSC52A99 (Convex and discrete geometry :: General convexity :: Miscellaneous)

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