It is well known that every compact convex set with non-empty interior $K\in\mathbb{R}^{n}$ has (many) circumscribing regular simplices. That is, there exists at least a simplex $\Delta$ such that $K\subseteq \Delta$ and $\Delta$ has a minimal volume.
I can not find references for infinite dimensional spaces.
Q1. Let $E$ be a locally convex space and $K\subset E$ a compact convex subset. Does it exist a Choquet or Bauer simplex $\Delta\subset E$ such that $K\subset\Delta$ ?
Q2. If yes, in which sense we have a ``minimal'' simplex to do the job.
