Category:Definitions/Extreme Points of Convex Sets

This category contains definitions related to Extreme Points of Convex Sets.
Related results can be found in Category:Extreme Points of Convex Sets.

Let $X$ be a vector space over $\R$.

Let $K$ be a convex subset of $X$.

Definition 1

We say that $a$ is an extreme point of $K$ if and only if:

whenever $a = t x + \paren {1 - t} y$ for $t \in \openint 0 1$, we have $x = y = a$.

Definition 2

We say that $a$ is an extreme point of $K$ if and only if:

$K \setminus \set a$ is convex.