Let $V$ be some `nice' vector space and let $T: V\to \mathbb{R}$ be a linear functional over $V$.

Define \begin{align} M= K \cap \bigcup_{i \in \mathbb{N} } \{ v \in V: T(v)=c_i \} \end{align} where $K$ is some compact and convex subset of $V$. Moreover, $K$ has at most $n$ extreme points.

That is, $M$ is an intersection of $K$ with countably many hyperplanes.

The question I have is, can we say something about the extreme points of $M$?

The general answer, I suspect, is that it is impossible to say something without extra assumptions on $T$. So, we would have to make some assumptions on $T$.

Some motivation: The following result can be shown when the intersection is finite.

Let $\tilde{M}=K \cap \bigcup_{i=1}^m \{ v \in V: T(v)=c_i \} $. Then, one can show, with little restriction on $T$, that the extreme points of $ \tilde{M}$ can be represented as a convex combination of at most $m$ extreme pints of $K$.