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I have 2 sets of 2D points, and each of the points P(x,y) satisfies these conditions

  1. $x \geq 0$
  2. $y \geq 0$
  3. $x + y \leq 1$

I am looking for a way to find out the similarity of the 2 sets.

In the "distance" Wikipedia page, the part "Distances between sets.." covers distances between sets of 1D points, such as Hausdorff distance. I would like to know if there are other distances between sets of 2D points.

Any comments are welcome!

Thanks Patrick

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2  
The Hausdorff distance has nothing to do with dimension. You simply have to use a metric for two dimensional space. This is not really MO level though. I think you should try math.stackexchange.com –  Michael Greinecker Oct 11 '10 at 7:46
1  
You should clarify: what do you mean exactly by similarity of the 2 sets? Is there any other property besides 1.2.3, that you left implicit, e.g. the sets are finite, closed...? –  Pietro Majer Oct 11 '10 at 7:53
    
Perhaps the following book is helpful: Encyclopedia of Distances springerlink.com/content/… –  Suvrit Oct 11 '10 at 9:22
3  
The "right" definition of similarity, and thus the distance, will depend of course on what application you have in mind. (For instance, will a rigid motion of your set give you something you would consider very similar?) Since you did not include any details, it is hard for MO users to be much more helpful than that. –  Thierry Zell Oct 11 '10 at 11:51
    
Note however that a pair $(x,y)$ satisfying your 1. 2. 3. is the same as a point in a closed triangle $T$; so your subsets are subsets of $T,$ and you can consider the associate Hausdorff distance between them (however, as usual, it is a genuine distance on closed sets, while on general subsets it's just a semi-distance). –  Pietro Majer Oct 11 '10 at 12:07

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