I struggle to understand the definition of "Extreme set" and "Extreme point".
Let $K$ a subset of a vector space $X$. A non-empty set $S \subset K$ is called an extreme set of $K$ if no point of $S$ is an internal point of any line whose whose end points are in $K$, except when both points are in $S$. Analytically, the condition can be expressed as follows: if $x, y \in K \;,\; 0 < t < 1$ and $$ (1-t)x + ty \in S $$ then $x \in S$ and $y \in S$.
Can anyone explain how does the analytical condition follow from the definition? For some reason it seems to me the definition itself is a bit twisted.
Also we have
The extreme points of $K$ are the extreme sets that consist of just one point. The set of all extreme points of $K$ will be denoted by $E(K)$.
What's an example set consisting of just a single point that isn't an extreme set?
The analytical description is just the parametrization of a line. For example, the line between $(1,1)$ and $(2,2)$ is given by $\{t(1,1)+(1-t)(2,2):t\in[0,1]\}$, Note that $t(1,1)+(1-t)(2,2)=(2-t,2-t)$ which indeed is the line between (1,1) and (2,2).
Now to understand the notion of extremeness it is important to understand extreme sets are extreme with regard to some other set $K$. Let's consider the square $K$ with corner point (0,0), (1,0), (1,1) and (0,1). The extreme points of the square are exactly the corner points (0,0), (1,0), (1,1) and (0,1). Extreme sets of $K$ are of course the extreme points and the extreme point sets, $K$ itself and the edges of $K$. (And unions of these sets.)
To proof this you can use the analytical definition of extreme sets. Take for example the edge between (1,0) and (1,1), this is the set $\{(1,\lambda):\lambda\in[0,1]\}$. Now let $x,y\in K$ such that there is some $t\in(0,1)$ such that $tx+(1-t)y=(1,\lambda)$ for some $\lambda\in[0,1]$. First of all note that $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$ with $x_{1},x_{2},y_{1},y_{2}\in[0,1]$. We find $$tx+(1-t)y=(tx_{1}+(1-t)y_{1},tx_{2}+(1-t)y_{2})=(1,\lambda).$$ So $1=tx_{1}+(1-t)y_{1}\leq t+t-1=1$ which implies that $x_{1}=y_{1}=1$ and thus $x,y\in\{(1,\lambda):\lambda\in[0,1]\}$ which therefore is an extreme set.
I hope this helped.