I think I have a proof of the following result:
Let $V$ be a separable real Banach space. Let $M \subset V^*$ be a nonempty convex subset of the unit ball in $V^*$ which is closed in the weak-$*$ topology. For any $\lambda \in M$, there exist an extreme point $\mu \in \mathcal{E}(M)$, another point $\nu \in M$, and some $t \in (0,1]$ such that $\lambda = t\mu + (1-t)\nu$.
It's immediate, from Krein-Milman plus the metrizability of the topology on $M$, that $M$ consists of limits of sequences of such convex combinations, but removing the limit took a bit of work. The sketch is to write $\lambda$ as a convergent sequence of finite convex combinations $\lambda_n$ of extreme points, then show that for at least one $\mu$, the $\limsup$ of the coefficient of $\mu$ in the expansion of $\lambda_n$ is positive. Extract a subsequence $\lambda_n = t_n \mu + \sum_{j} c_{n,j} \mu_{n,j}$ for some other extreme points $\mu_{n,j}$, where $0 < t_n \to t$, then take another subsequence to make $(1-t_n)^{-1} \sum_{j} c_{n,j} \mu_{n,j}$ converge to some $\nu \in K$ (using the separable Banach stuff), and add things up.
My questions:
- This seems like such a geometrically obvious fact, but the proof I have, with the details $\TeX$'d out, is more than a page. Is there an obvious, slicker proof that I've missed?
- Can I relax the Banach assumption to replace $V^*$ with a locally convex Hausdorff topological vector space?
