Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Staff Emeritus
    Science Advisor
    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook