2

The Hausdorff distance is defined for non-empty sets. What would be a reasonable generalization of the definition for the case when one of the sets is empty, if the generalized distance should remain a pseudo-metric?

Initially I thought of 0, but it violates the triangle inequality since it would imply that for all $X$,$Y$:

$$H(X,Y)\leq H(X,\phi)+H(\phi,Y)=0$$

which is of course false.

$\infty$ does satisfy the triangle inequality, but requires a special treatment for $H(\phi,\phi)=0$.

Another option, which I think doesn't violate any axiom of pseudo-metric, is to define $H(X,\phi)$ as the Hausdorff distance between $X$ and a constant set, say, the origin.

Does this generalization make sense?

Erel Segal-Halevi
  • 10,555

1 Answers1

7

The most natural thing to do is just treat the same definition for where you allow the empty set. You get that $H(X,\emptyset)=H(\emptyset, X)=\infty $ for all non-empty $X$ and that $H(\emptyset,\emptyset)=0$. It's not a special case or anything, just the definition.

The thing is we don't really care about the distance from the empty set. It's not that it matters much, but it's kind of irrelevant to what the Hausdorff distance measures.

Ittay Weiss
  • 79,840
  • 7
  • 141
  • 236