Let $X$ be a Banach space. Assume $K$ is compact, and convex subset of $X$.
My first question: Do we have $K = \text{conv} \; (ext K)$? where $ext K$ denotes the set of all extrem points of $K$.
My second and main question: Assume $K \subset X^*$ which is convex, norm bounded and weak* compact. Do we have $K = \text{conv} \; (ext K)$?
I think answer of the first question is NO. Because by Krein-Miliman Theorem we only can say $K $ is the closure of $ \text{conv} \; (ext K)$ and I dont think being Banach is a bonus here.