0
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

Hausdorff distance is what you are looking for.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.