I have an issue with a definition in Rudin's Functional Analysis in the paragraph regarding the Krein-Milman Theorem.
"Let $K$ be a subset of a vector space $X$. A nonempty set $S$ in $K$ is called an extreme set if no point of $S$ is an internal point of a line interval whose end points are in $K$ but not in $S$. Analytically, the condition can be expressed as follows: if $x$ and $y$ are in $K$, if $t$ is in $(0, 1)$, and if $tx + (1 - t)y$ is in $S$, then $x$ and $y$ are in $S$. The extreme points of $K$ are the extreme sets that consist of just one point."
For this condition to be equivalent to the definition, one should replace the conclusion by: "$\dots$ then $x$ is in $S$ or $y$ is in $S$." It turns out that this is indeed equivalent when $S$ consists of a single point, but not in general.
So my question is: what is the good definition for an extreme set?
The second sentence is convoluted and, to me, hard to understand. The third sentence conforms to several other texts. If you want a fancier definition you can use the following:
Let $\mathsf{con}$ denote the convex hull operator. For $S \subseteq K$, the set $S$ is an extreme subset of $K$ if and only if for all $D \subseteq K$ we have $S \cap \mathsf{con}(D) = S \cap \mathsf{con}(S \cap D)$. If you want to, you only have to consider finite subsets $D$.