# Smallest simplex

(Difference between revisions)
 Revision as of 04:15, 17 February 2010 (edit)m (Reverted edits by Ranjelin (Talk); changed back to last version by 83.234.14.118)← Previous diff Revision as of 04:16, 17 February 2010 (edit) (undo)Next diff → Line 1: Line 1: - zWQhZt augfhajsjsvj, [url=http://ssxklvxsdjvp.com/]ssxklvxsdjvp[/url], [link=http://gdyhpofdhosn.com/]gdyhpofdhosn[/link], http://kzlosdhopeta.com/ + 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? + + ==question== + ''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?'' + + ==Reply== + Yes, if a simplex in n-space having full dimension, then it has (n+1) vertices. But here we allow for sub-dimensional simplexes. I don't think measuring the volume of them is pointless if we constrain our focus to the sub affine set where the simplex resides. + This is just like we can measure the area of a triangle in a 3 dimensional space. + --[[User:Flyshcool|Flyshcool]] 14:30, 1 July 2008 [GMT+8]

## Revision as of 04:16, 17 February 2010

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?

## question

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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