# Existence of circumscribing simplex in infinite dimensional space

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.

• if you have countable dimension then perhaps you can talk about minimising the average edge length of $\Delta$ ? – Dima Pasechnik Nov 7 '17 at 12:27
• I don't understand the introductory statements. What would be the minimum volume $\Delta$ for the unit ball of $\mathbb R^n$ ? – Christian Remling Nov 7 '17 at 20:23