1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ if you have countable dimension then perhaps you can talk about minimising the average edge length of $\Delta$ ? $\endgroup$ Nov 7, 2017 at 12:27
  • $\begingroup$ I don't understand the introductory statements. What would be the minimum volume $\Delta$ for the unit ball of $\mathbb R^n$ ? $\endgroup$ Nov 7, 2017 at 20:23

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.