Talk:Simplex
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Hello, AxelBoldt! I see that you replaced "linearly independent points" with "points in general position in some Euclidean space". I'm not sure what you mean by "in general position" - is this a technical expression? Is it more accurate than saying the points have to be linearly independent? Thanks for any clarification you can give in this matter! -- Oliver Pereira 23:11 Nov 23, 2002 (UTC)
"Linearly independent" is technically incorrect: for example, the points (1,1), (1,0), (0,1) in R^2 are linearly dependent, but they span a 2-simplex. If you require linear independence, there won't be any 2-simplices in R^2.
There is probably a technical definition of "in general position", but I don't know it. Typically, the term is used to describe points that don't satisfy "more equations than necessary"; for instance if you have four points that all lie on a circle, or three points that all lie on a line, then they wouldn't be in general position.
It's probably not the best term to use here. Maybe we should go with the formally correct "affinely independent", which precisely means what we want: any m-plane contains at most m+1 of the points. AxelBoldt 19:46 Nov 24, 2002 (UTC)
- Oh, of course! Silly me. I clearly wasn't thinking straight about the linear independence thing. It was late, after all. :) Thanks for clearing up my confusion. -- Oliver Pereira 21:01 Nov 24, 2002 (UTC)
I don't get this. Shouldn't it say that <math>\sum_i t_i<math> is less than or equal, not equal to 1 in the geometric definition?
- No, the n-simplex is given as a subset of Rn+1 not Rn. It must therefore lie in an n-dimensional (affine) hyperplane. -- Fropuff 20:13, 2005 May 23 (UTC)