Smallest simplex

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I am a PhD. candidate student in Tsinghua University, China. I think this is an open problem in my field. That is:

How to find the smallest simplex which can enclose a bunch of given points in a high dimensional space (under the following two assumptions:)?

  • (1) The number of the vertexes of the simplex is known, say n;
  • (2) The number of the vertexes of the simplex is unknown.

To measure how small the simplex is, we can use the volume of the simplex.

The question is: can this problem be cast into a convex optimization?

Doesn't a simplex in n-space always have (n+1) vertices (http://en.wikipedia.org/wiki/Simplex)? Or would you want to allow for sub-dimensional simplices? But then, measuring the volume is quite pointless for all but the full-dimensional ones. I'm probably misunderstanding something, perhaps you can clarify this?

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