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Smallest Set Containing All n-point Sets

  1. Jul 5, 2010 #1
    I've been thinking about this. Suppose you have an n-point set P in Rm which has the property that for any two points x, y in P, ||x - y|| < 2. If we fix n, what can we say about the smallest set S in Rm that contains P, allowing for both translations and orthogonal transformations of S?

    If we start in R2, the answer is not what you'd expect! As illustrated in this picture, P must be able to fit inside of any 2 by 2 square after only translation:
    attachment.php?attachmentid=26839&stc=1&d=1278316407.png
    That is, if I were to have a square hoola-hoop, I could swing it around P. However, for n > 2, the intersection of all those squares would not necessarily fit inside a circle of diameter 2.

    For n=2, the smallest set would be something like [0,1] U {2}.
    For n=3, I think the smallest set would be {x = (x1, x2) : ||x - (0,1)|| < 2, x1 < 0, x2 < 0} U {x = (x2,x2) : x1 = 0, -1 < x2 < 1}

    Any thoughts?
     

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  3. Jul 5, 2010 #2

    radou

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    I'm not sure what exactly you mean by "translations" and "orthogonal transformation".

    If you have an n-point subset P of R^m, take max{||x - y||, x, y from P} = diam(P) = r = d(x0, y0) (for some x0, y0 in P), and then the closed ball B(x0, r) around x with radius r contains the set P.
     
  4. Jul 5, 2010 #3
    Yes, but that's not the smallest. :wink:
    Take the intersection of the balls about x0 and y0. This would also necessarily contain P.

    About the "translations and orthogonal transformations" business: I'm looking for a closed form for a set S which must "be able to contain" P. By "be able to contain", I mean that for any P, we ought to find some transformation T (involving only rotations, reflections, and translations) such that P is contained in T(S).
     
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