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N distinct points in R^n-1 or higher one can be separated in an equal distance(>0)?

  1. Apr 24, 2012 #1
    Is there a theorem that states that n distinct points in R^n-1 or higher one can be separated in an equal distance as the distance is greater than 0?

    We know that 4 distinct points in R^2 cannot be positioned in an equal distance>0 but in R^3 it is possible as a pyramid shape.

    If there is such a theorem, could you give me reference?

    I've posted this on this area because it seems it is a problem of solving a system of equations.

    Cheers
     
  2. jcsd
  3. Apr 24, 2012 #2
    Re: n distinct points in R^n-1 or higher one can be separated in an equal distance(>0

    I do not know the reference, but it is a simple fact which is easy to prove:

    Take in [itex]\mathbb R^n[/itex] n points (1, 0, ...,0), (0, 1, 0, ..., 0), ... (0, 0, ... , 0, 1) (the standard unit vectors). Then the distance between any 2 points is the same ([itex]\sqrt 2[/itex]). Now take the (n-1) dimensional hyperplane through these points, and you get the points positioned in [itex]\mathbb R^{n-1}[/itex].
     
  4. Apr 26, 2012 #3
    Re: n distinct points in R^n-1 or higher one can be separated in an equal distance(>0

    Oh yeah... Thank you indeed!
     
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