<p><a href="http://en.wikipedia.org/wiki/Metric_space#Distance_between_points_and_sets.3B_Hausdorff_distance_and_Gromov_metric" rel="nofollow">Hausdorff distance</a> is what you are looking for.</p>

CC BY-SA 3.0
2014-02-08T09:09:37.643Z
2014-02-08T09:09:37.643Z
Dibyayan
<p>Given a set of points $p_1, p_2, ..., p_n$ and a collection of sets from $n$ points $S_1, S_2,...,S_m$. The pairwise distance between any three points $(p_1, p_2, p_3)$ satisfies triangle inequality $d(p_1,p_2) + d(p_2, p_3) \geq d(p_1, p_3)$. How to define the distance between sets such that the pairwise distance between any three sets $S_1, S_2, S_3$ satisfies triangle inequality $d(S_1,S_2) + d(S_2, S_3) \geq d(S_1, S_3)$. 
For example, when every set is $\textbf{pairwise disjoint}$ and has the $\textbf{same cardinality}$, we can define the distance between any two set to be the sum of the distances between elements of the two sets $$d(S_1, S_2) = \sum_{x \in S_1, y \in S_2} d(x,y)$$. we can show that the triangle inequality is satisfied for any three sets. </p>

<metrics><metric-spaces><computational-geometry>
Define metric for systems of set of points
Hung Le
# Define metric for systems of set of points

(Asked by: username_0 on 2014-02-08)

Given a set of points $p\_1, p\_2, ..., p\_n$ and a collection of sets from $n$ points $S\_1, S\_2,...,S\_m$. The pairwise distance between any three points $(p\_1, p\_2, p\_3)$ satisfies triangle inequality $d(p\_1,p\_2) + d(p\_2, p\_3) \geq d(p\_1, p\_3)$. How to define the distance between sets such that the pairwise distance between any three sets $S\_1, S\_2, S\_3$ satisfies triangle inequality $d(S\_1,S\_2) + d(S\_2, S\_3) \geq d(S\_1, S\_3)$.
For example, when every set is $\textbf{pairwise disjoint}$ and has the $\textbf{same cardinality}$, we can define the distance between any two set to be the sum of the distances between elements of the two sets $$d(S\_1, S\_2) = \sum\_{x \in S\_1, y \in S\_2} d(x,y)$$. we can show that the triangle inequality is satisfied for any three sets.

**Tags:** metrics, metric-spaces, computational-geometry


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## 1 answers

### Answer by username_1 on 2014-02-08. Score: 4

[Hausdorff distance](http://en.wikipedia.org/wiki/Metric_space#Distance_between_points_and_sets.3B_Hausdorff_distance_and_Gromov_metric) is what you are looking for.


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