Finite Metric Spaces

  • Thread starter Dragonfall
  • Start date
  • #1
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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)?
 

Answers and Replies

  • #2
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Can you do a set with 4 points?
 
  • #3
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What do you mean?
 
  • #4
Hurkyl
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Science Advisor
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"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?)
 
  • #5
1,030
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OK, it seems this one

o
\
o ---- 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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