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Finite Metric Spaces

  1. Sep 10, 2009 #1
    Is every finite metric space imbeddable in a manifold?

    That is, for every finite metric space (X,d), does there exist some manifold such that there are |X| many points on it which is isometric (with the length of geodesics as metric) to (X,d)?
  2. jcsd
  3. Sep 10, 2009 #2
    Can you do a set with 4 points?
  4. Sep 10, 2009 #3
    What do you mean?
  5. Sep 10, 2009 #4


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    Staff Emeritus
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    Gold Member

    "Every finite metric space" is a tall order. Try starting with embedding "every metric space with 4 points" into Riemannian manifolds. (Actually, why not start with 0, 1, 2, 3, and then 4?)

    (P.S. I assume you mean "connected and Riemannian" manifolds?)
  6. Sep 10, 2009 #5
    OK, it seems this one

    o ---- o

    where the metric is the number of steps between nodes, can't be done. But I can't prove it conclusively.
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