Smallest simplex

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How to find the smallest simplex which can enclose a bunch of given points in a high dimensional space (under the following two assumptions:)?
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;
*(1) The number of the vertexes of the simplex is known, say n;
-
*(2) The number of the vertexes of the simplex is unknown;
+
*(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.
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?
The question is: can this problem be cast into a convex optimization?

Revision as of 01:59, 12 June 2008

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?

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